MAGMA commands to produce the Molien series:

K:=QuadraticField(2);
M:=MatrixAlgebra(K,16);
V:=VectorSpace(GF(2),4);
VV:=[];
for v in V do Append(~VV,v); end for;

q:=M ! 0;
for i in [1..16] do q[i][i]:=(-1)^(Integers() ! VV[i][1]); end for;

g:=GL(4,GF(2));
gg:= Setseq(Generators(g));

g1:=gg[1];
g2:=gg[2];

r1:=M ! 0;
r2:=M ! 0;
for i in [1..16] do v:=VV[i]; w:=v*g1; p:=Position(VV,w); r1[i][p]:=1;
end for;
for i in [1..16] do v:=VV[i]; w:=v*g2; p:=Position(VV,w); r2[i][p]:=1;
end for;

h:=M ! [
1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1
];
h:=h/K.1;

//for i in [1..16] do for j in [1..16] do h[i][j]:=(-1)^(Integers() ! (VV[i][1]*VV[j][1])); end for; end for;


G:=MatrixGroup<16,K|q,r1,r2,h>;

N:=NormalClosure(G,sub<G|q>);
 U:=sub<G | h^(-1)*r2*h,r1,r2,q>;
P,Q,R:=CosetAction(G,U);
R eq N;
CC:=ConjugacyClasses(Q);
NN:=[];
for n in N do Append(~NN,n); end for;

RR<t>:=RationalFunctionField(K); Mol:=RR ! 0;
MRR:=MatrixRing(RR,16);
for i in [1..#CC] do 
print i;
c:=CC[i];
z:=c[3]@@P;
l:=c[2];
s:=RR ! 0;
for n in NN do 
  s:=s+ Determinant(MRR ! (1-t*MRR ! (z*n)))^(-1);
end for;
Mol:=Mol + l*s;
end for;

PrintFile("Molgen4",Mol,"Magma");
quit;



The results:

P<t>:=RationalFunctionField(Rationals());
molgen4:=1/178362777600*(178362777600*t^456 - 178362777600*t^454 - 356725555200*t^450 + 
    356725555200*t^448 - 178362777600*t^446 + 535088332800*t^444 - 
    356725555200*t^442 + 535088332800*t^440 - 713451110400*t^438 + 
    891813888000*t^436 - 713451110400*t^434 + 1605264998400*t^432 - 
    1605264998400*t^430 + 1961990553600*t^428 - 2140353331200*t^426 + 
    4637432217600*t^424 - 2497078886400*t^422 + 8204687769600*t^420 - 
    3388892774400*t^418 + 14090659430400*t^416 - 2853804441600*t^414 + 
    24970788864000*t^412 - 178362777600*t^410 + 46552684953600*t^408 + 
    7847962214400*t^406 + 80976701030400*t^404 + 27111142195200*t^402 + 
    145008938188800*t^400 + 68491306598400*t^398 + 252740055859200*t^396 + 
    144117124300800*t^394 + 423076508467200*t^392 + 276105579724800*t^390 + 
    694187930419200*t^388 + 498702326169600*t^386 + 1111200104448000*t^384 + 
    849898635264000*t^382 + 1718168636620800*t^380 + 1376960643072000*t^378 + 
    2600886022963200*t^376 + 2162648678400000*t^374 + 3836940071731200*t^372 + 
    3266357546188800*t^370 + 5514263632281600*t^368 + 4781549341900800*t^366 + 
    7763061532262400*t^364 + 6832364558745600*t^362 + 10695702321561600*t^360 + 
    9506022594969600*t^358 + 14440072111718400*t^356 + 12918637618790400*t^354 +
    19146709087027200*t^352 + 17238227366707200*t^350 + 24943142633472000*t^348 
    + 22538098940313600*t^346 + 31961004480921600*t^344 + 
    28926696908390400*t^342 + 40334601800908800*t^340 + 36587556569088000*t^338 
    + 50161142307225600*t^336 + 45520677922406400*t^334 + 
    61523207965900800*t^332 + 55802043511603200*t^330 + 74496067869081600*t^328 
    + 67562036527104000*t^326 + 89083110909542400*t^324 + 
    80714329384550400*t^322 + 105296822481715200*t^320 + 95217185193984000*t^318
    + 123070316544000000*t^316 + 111182794142515200*t^314 + 
    142339204133683200*t^312 + 128323278707097600*t^310 + 
    162949023085363200*t^308 + 146563548158361600*t^306 + 
    184778486710272000*t^304 + 165885587855769600*t^302 + 
    207574141501440000*t^300 + 185934277509120000*t^298 + 
    231185627637350400*t^296 + 206485415106969600*t^294 + 
    255287076323328000*t^292 + 227562009447628800*t^290 + 
    279681218327347200*t^288 + 248639138876620800*t^286 + 
    304025775479193600*t^284 + 269532911370240000*t^282 + 
    328116165672960000*t^280 + 290204265480192000*t^278 + 
    351571406015692800*t^276 + 310178042766950400*t^274 + 
    374254692256972800*t^272 + 329202930076876800*t^270 + 
    395771485932748800*t^268 + 347388085429862400*t^266 + 
    416006207963136000*t^264 + 364178800587571200*t^262 + 
    434629065572352000*t^260 + 379446475987353600*t^258 + 
    451563006040473600*t^256 + 393329342781849600*t^254 + 
    466485906191155200*t^252 + 405375072967065600*t^250 + 
    479448599416012800*t^248 + 415457385696460800*t^246 + 
    490132173068697600*t^244 + 423857559070310400*t^242 + 
    498601194474700800*t^240 + 430114703671296000*t^238 + 
    504644125379788800*t^236 + 434196714199449600*t^234 + 
    508354427879424000*t^232 + 436426605645004800*t^230 + 
    509517888277708800*t^228 + 436426605645004800*t^226 + 
    508354427879424000*t^224 + 434196714199449600*t^222 + 
    504644125379788800*t^220 + 430114703671296000*t^218 + 
    498601194474700800*t^216 + 423857559070310400*t^214 + 
    490132173068697600*t^212 + 415457385696460800*t^210 + 
    479448599416012800*t^208 + 405375072967065600*t^206 + 
    466485906191155200*t^204 + 393329342781849600*t^202 + 
    451563006040473600*t^200 + 379446475987353600*t^198 + 
    434629065572352000*t^196 + 364178800587571200*t^194 + 
    416006207963136000*t^192 + 347388085429862400*t^190 + 
    395771485932748800*t^188 + 329202930076876800*t^186 + 
    374254692256972800*t^184 + 310178042766950400*t^182 + 
    351571406015692800*t^180 + 290204265480192000*t^178 + 
    328116165672960000*t^176 + 269532911370240000*t^174 + 
    304025775479193600*t^172 + 248639138876620800*t^170 + 
    279681218327347200*t^168 + 227562009447628800*t^166 + 
    255287076323328000*t^164 + 206485415106969600*t^162 + 
    231185627637350400*t^160 + 185934277509120000*t^158 + 
    207574141501440000*t^156 + 165885587855769600*t^154 + 
    184778486710272000*t^152 + 146563548158361600*t^150 + 
    162949023085363200*t^148 + 128323278707097600*t^146 + 
    142339204133683200*t^144 + 111182794142515200*t^142 + 
    123070316544000000*t^140 + 95217185193984000*t^138 + 
    105296822481715200*t^136 + 80714329384550400*t^134 + 89083110909542400*t^132
    + 67562036527104000*t^130 + 74496067869081600*t^128 + 
    55802043511603200*t^126 + 61523207965900800*t^124 + 45520677922406400*t^122 
    + 50161142307225600*t^120 + 36587556569088000*t^118 + 
    40334601800908800*t^116 + 28926696908390400*t^114 + 31961004480921600*t^112 
    + 22538098940313600*t^110 + 24943142633472000*t^108 + 
    17238227366707200*t^106 + 19146709087027200*t^104 + 12918637618790400*t^102 
    + 14440072111718400*t^100 + 9506022594969600*t^98 + 10695702321561600*t^96 +
    6832364558745600*t^94 + 7763061532262400*t^92 + 4781549341900800*t^90 + 
    5514263632281600*t^88 + 3266357546188800*t^86 + 3836940071731200*t^84 + 
    2162648678400000*t^82 + 2600886022963200*t^80 + 1376960643072000*t^78 + 
    1718168636620800*t^76 + 849898635264000*t^74 + 1111200104448000*t^72 + 
    498702326169600*t^70 + 694187930419200*t^68 + 276105579724800*t^66 + 
    423076508467200*t^64 + 144117124300800*t^62 + 252740055859200*t^60 + 
    68491306598400*t^58 + 145008938188800*t^56 + 27111142195200*t^54 + 
    80976701030400*t^52 + 7847962214400*t^50 + 46552684953600*t^48 - 
    178362777600*t^46 + 24970788864000*t^44 - 2853804441600*t^42 + 
    14090659430400*t^40 - 3388892774400*t^38 + 8204687769600*t^36 - 
    2497078886400*t^34 + 4637432217600*t^32 - 2140353331200*t^30 + 
    1961990553600*t^28 - 1605264998400*t^26 + 1605264998400*t^24 - 
    713451110400*t^22 + 891813888000*t^20 - 713451110400*t^18 + 
    535088332800*t^16 - 356725555200*t^14 + 535088332800*t^12 - 
    178362777600*t^10 + 356725555200*t^8 - 356725555200*t^6 - 178362777600*t^2 +
    178362777600)/(t^472 - 2*t^470 + t^468 - 2*t^466 + 3*t^464 - t^462 + 2*t^460
    - 2*t^458 - t^454 + 2*t^452 - 2*t^448 + t^446 - 3*t^444 + 3*t^442 - t^440 + 
    4*t^438 - 3*t^436 + t^434 - 4*t^432 + 5*t^430 - 3*t^428 + 5*t^426 - 6*t^424 
    + 2*t^422 - 3*t^420 + 6*t^418 - 4*t^416 + 4*t^414 - 6*t^412 + 3*t^410 - 
    t^408 + 2*t^406 - t^404 - 2*t^402 + 2*t^400 - t^398 + 4*t^396 - 2*t^394 + 
    3*t^392 - 9*t^390 + 6*t^388 - 4*t^386 + 9*t^384 - 10*t^382 + 6*t^380 - 
    11*t^378 + 14*t^376 - 7*t^374 + 12*t^372 - 15*t^370 + 8*t^368 - 11*t^366 + 
    12*t^364 - 6*t^362 + 7*t^360 - 8*t^358 + 2*t^356 - t^354 + 2*t^352 + 6*t^350
    - 5*t^348 + t^346 - 10*t^344 + 11*t^342 - 8*t^340 + 10*t^338 - 11*t^336 + 
    11*t^334 - 11*t^332 + 14*t^330 - 13*t^328 + 13*t^326 - 14*t^324 + 15*t^322 -
    16*t^320 + 11*t^318 - 12*t^316 + 11*t^314 - 8*t^312 + 10*t^310 - 6*t^308 + 
    3*t^306 - 3*t^304 - t^302 + 3*t^300 - t^298 + 4*t^296 - 9*t^294 + 6*t^292 - 
    4*t^290 + 10*t^288 - 10*t^286 + 10*t^284 - 13*t^282 + 10*t^280 - 11*t^278 + 
    11*t^276 - 13*t^274 + 14*t^272 - 9*t^270 + 11*t^268 - 10*t^266 + 6*t^264 - 
    4*t^262 + 3*t^260 + t^258 - 3*t^256 - t^254 - 6*t^252 + 5*t^250 - t^248 + 
    12*t^246 - 8*t^244 + 2*t^242 - 10*t^240 + 9*t^238 - 4*t^236 + 9*t^234 - 
    10*t^232 + 2*t^230 - 8*t^228 + 12*t^226 - t^224 + 5*t^222 - 6*t^220 - t^218 
    - 3*t^216 + t^214 + 3*t^212 - 4*t^210 + 6*t^208 - 10*t^206 + 11*t^204 - 
    9*t^202 + 14*t^200 - 13*t^198 + 11*t^196 - 11*t^194 + 10*t^192 - 13*t^190 + 
    10*t^188 - 10*t^186 + 10*t^184 - 4*t^182 + 6*t^180 - 9*t^178 + 4*t^176 - 
    t^174 + 3*t^172 - t^170 - 3*t^168 + 3*t^166 - 6*t^164 + 10*t^162 - 8*t^160 +
    11*t^158 - 12*t^156 + 11*t^154 - 16*t^152 + 15*t^150 - 14*t^148 + 13*t^146 -
    13*t^144 + 14*t^142 - 11*t^140 + 11*t^138 - 11*t^136 + 10*t^134 - 8*t^132 + 
    11*t^130 - 10*t^128 + t^126 - 5*t^124 + 6*t^122 + 2*t^120 - t^118 + 2*t^116 
    - 8*t^114 + 7*t^112 - 6*t^110 + 12*t^108 - 11*t^106 + 8*t^104 - 15*t^102 + 
    12*t^100 - 7*t^98 + 14*t^96 - 11*t^94 + 6*t^92 - 10*t^90 + 9*t^88 - 4*t^86 +
    6*t^84 - 9*t^82 + 3*t^80 - 2*t^78 + 4*t^76 - t^74 + 2*t^72 - 2*t^70 - t^68 +
    2*t^66 - t^64 + 3*t^62 - 6*t^60 + 4*t^58 - 4*t^56 + 6*t^54 - 3*t^52 + 2*t^50
    - 6*t^48 + 5*t^46 - 3*t^44 + 5*t^42 - 4*t^40 + t^38 - 3*t^36 + 4*t^34 - t^32
    + 3*t^30 - 3*t^28 + t^26 - 2*t^24 + 2*t^20 - t^18 - 2*t^14 + 2*t^12 - t^10 +
    3*t^8 - 2*t^6 + t^4 - 2*t^2 + 1);
D:=Denominator(molgen4);
N:=Numerator(molgen4);
Q<x>:=PolynomialRing(Rationals());
D:=Q ! D;
N:=Q ! N;
P<t>:=PowerSeriesRing(Rationals(),100);
mm:=P ! molgen4;
mm;

Factorization(Q! (D*(1+x^2)*(1+x^4)*(1+x^8)*(1+x^6)^2*(1+x^12)*(1+x^10)/((1-\
1-x^12)*(1-x^24)*(1-x^8)*(1-x^20)*(1-x^14)*(1-x^16)^2*(1-x^18)*(1-x^24)*\
(1-x^30)*(1-x^40)*(1-x^48)*(1-x^56)*(1-x^72)*(1-x^120))));

 N*(1+x^2)*(1+x^4)*(1+x^8)*(1+x^6)^2*(1+x^12)*(1+x^10);


Latex version of Molien series:

The Molien series of ${\cal C}_4$ is 
\beql{MSC4}
\MS ({\cal C}_4) = \frac{f(t^2)+t^{504}f(t^{-2})}{g(t^2)} 
\eeql
where
$$ 
\begin{array}{l}
f(t) := 
1 + 1 t^{ 9 }+ 3 t^{ 10 }+ 4 t^{ 11 }+ 8 t^{ 12 }+ 9 t^{ 13 }+ 
16 t^{ 14 }+ 21 t^{ 15 }+ 36 t^{ 16 }+ 
 45 t^{ 17 }+ \\  78 t^{ 18 }+ 107 t^{ 19 }+ 
173 t^{ 20 }+ 254 t^{ 21 }+ 393 t^{ 22 }+ 570 t^{ 23 }+ 883 t^{ 24 }+ 1271 t^{ 25 }+ 
1895 t^{ 26 }+ \\ 2728 t^{ 27 }+  3974 t^{ 28 }+ 5632 t^{ 29 }+ 8072 t^{ 30 }+ 11266 t^{ 31 }+ 
15784 t^{ 32 }+ 21748 t^{ 33 }+ 29885 t^{ 34 }+ 
\\ 40494 t^{ 35 }+ 54686 t^{ 36 }+ 72906 t^{ 37 }+ 
96703 t^{ 38 }+ 126951 t^{ 39 }+ 165644 t^{ 40 }+ 214119 t^{ 41 }+ \\
275083 t^{ 42 }+ 350401 t^{ 43 }+ 
443551 t^{ 44 }+ 557196 t^{ 45 }+ 695660 t^{ 46 }+ 862411 t^{ 47 }+ 1062746 t^{ 48 }+ \\ 1301205 t^{ 49 }+ 
1584011 t^{ 50 }+ 1916678 t^{ 51 }+ 2306665 t^{ 52 }+ 2760470 t^{ 53 }+ 3286285 t^{ 54 }+ \\ 3892204 t^{ 55 }+ 
4586981 t^{ 56 }+ 5379348 t^{ 57 }+ 6279586 t^{ 58 }+ 7296922 t^{ 59 }+  8441388 t^{ 60 }+  \\
9724111 t^{ 61 }+ 
11154744 t^{ 62 }+ 12743916 t^{ 63 }+ 14502733 t^{ 64 }+ 16441119 t^{ 65 }+ 18568345 t^{ 66 }+ \\ 20896306 t^{ 67 }+ 
23432240 t^{ 68 }+ 26185639 t^{ 69 }+ 29165245 t^{ 70 }+ 32378239 t^{ 71 }+ 35829251 t^{ 72 }+ \\  39527507 t^{ 73 }+ 
43474073 t^{ 74 }+ 47673490 t^{ 75 }+ 52128673 t^{ 76 }+ 56840104 t^{ 77 }+ 61804658 t^{ 78 }+ \\67025522 t^{ 79 }+ 
72494722 t^{ 80 }+ 78208571 t^{ 81 }+ 84162405 t^{ 82 }+ 90347388 t^{ 83 }+ 96751384 t^{ 84 }+ \\103369746 t^{ 85 }+ 
110185042 t^{ 86 }+ 117184592 t^{ 87 }+ 124356711 t^{ 88 }+ 131683108 t^{ 89 }+\\ 139144151 t^{ 90 }+  146728446 t^{ 91 }+ 
154411344 t^{ 92 }+ 162173029 t^{ 93 }+ 169997892 t^{ 94 }+ \\177860837 t^{ 95 }+  185738591 t^{ 96 }+ 193616190 t^{ 97 }+ 
201465931 t^{ 98 }+ 209264797 t^{ 99 }+\\ 216997649 t^{ 100 }+  224636598 t^{ 101 }+ 232159221 t^{ 102 }+ 239551259 t^{ 103 }+ 
246786164 t^{ 104 }+\\ 253842006 t^{ 105 }+  260707824 t^{ 106 }+ 267356983 t^{ 107 }+ 273771501 t^{ 108 }+ 279941236 t^{ 109 }+  \\
285843610 t^{ 110 }+ 291461043 t^{ 111 }+ 296788860 t^{ 112 }+ 301804313 t^{ 113 }+ 306495503 t^{ 114 }+ \\ 310858074 t^{ 115 }+ 
314874748 t^{ 116 }+ 318533677 t^{ 117 }+ 321836735 t^{ 118 }+ 324765975 t^{ 119 }+ \\ 327315881 t^{ 120 }+   329488076 t^{ 121 }+ 
 331270427 t^{ 122 }+ 332656955 t^{ 123 }+\\ 333655428 t^{ 124 }+  334252987 t^{ 125 }+ 167224924 t^{ 126 }  
\end{array} 
$$
and 
$$ \begin{array}{l} g(t) = 
(1-t)(1-t^4)(1-t^6)(1-t^7)(1-t^8)^2(1-t^9)(1-t^{10})(1-t^{12})^2 \\
(1-t^{15})(1-t^{20}) 
(1-t^{24})(1-t^{28}) 
(1-t^{36})(1-t^{60}) 
\end{array}
$$
The first terms of $\MS({\cal C}_4) $ are 
$$
\begin{array}{l}
1 + t^{2} + t^{4} + t^{6} + 2 t^{8} + 2 t^{10} + 3 t^{12} + 4 t^{14} + 7 t^{16} + 9 t^{18} + 
    14 t^{20} + 19 t^{22} + \\ 33 t^{24} + 45 t^{26} + 69 t^{28} + 100 t^{30} + 159 t^{32} + 
    228 t^{34} + 355 t^{36} + 526 t^{38} + 815 t^{40} + \\ 1215 t^{42} + 1861 t^{44} + 
    2777 t^{46} + 4240 t^{48} + 6318 t^{50} + 9508 t^{52} + 14107 t^{54} + 21034 t^{56} + \\
    30927 t^{58} + 45564 t^{60} + 66382 t^{62} + 96585 t^{64} + 139194 t^{66} + 
    200003 t^{68} + \\ 284914 t^{70} + 404367 t^{72} + 569372 t^{74} + 798097 t^{76} + 
    1110803 t^{78} + 1538672 t^{80} + \\ 2117206 t^{82} + 2899191 t^{84} + 3945647 t^{86} + 
    5343821 t^{88} + 7196118 t^{90} + \\ 9644509 t^{92} + 12856864 t^{94} + 17060650 t^{96} + 
    22525587 t^{98} + O(t^{100})
\end{array}
$$