Second cohomology group for trivial group action of direct product of D8 and Z2 on Z2

Construction of the cohomology group
The cohomology group can be constucted using the GAP functions CyclicGroup, DirectProduct, TwoCohomology, TrivialGModule, GF.

gap> G := DirectProduct(DihedralGroup(8),CyclicGroup(2));; gap> A := TrivialGModule(G,GF(2));; gap> T := TwoCohomology(G,A); rec( group := , module := rec( field := GF(2), isMTXModule := true, dimension := 1, generators := [ , , ,  ] ), collector := rec( relators := [ [ 0 ], [ [ 2, 1, 3, 1 ], [ 3, 1 ] ], [ [ 3, 1 ], [ 3, 1 ], 0 ], [ [ 4, 1 ], [ 4, 1 ], [ 4, 1 ], 0 ] ],     orders := [ 2, 2, 2, 2 ], wstack := [ [ 1, 1 ], [ 2, 1, 3, 1 ] ], estack := [ ], pstack := [ 3, 5 ], cstack := [ 1, 1 ], mstack := [ 0, 0 ], list := [ 0, 0, 0, 0 ], module := [ , , ,  ], mone := , mzero := , avoid := [ ], unavoidable := [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] ), cohom :=  -> ( GF(2)^6 )>, presentation := rec( group := , relators := [ f1^2, f1^-1*f2*f1*f3^-1*f2^-1, f2^2*f3^-1, f1^-1*f3*f1*f3^-1, f2^-1*f3*f2*f3^-1, f3^2, f1^-1*f4*f1*f4^-1, f2^-1*f4*f2*f4^-1, f3^-1*f4*f3*f4^-1, f4^2 ] ) )

Construction of extensions
The extensions can be constructed using the additional command Extensions and the hand-coded command FrequencySort.

gap> G := DirectProduct(DihedralGroup(8),CyclicGroup(2));; gap> A := TrivialGModule(G,GF(2));; gap> L := Extensions(G,A);; gap> FrequencySort(List(L,IdGroup)); [ [ [ 32, 22 ], 2 ], [ [ 32, 23 ], 1 ], [ [ 32, 25 ], 4 ], [ [ 32, 27 ], 4 ], [ [ 32, 28 ], 8 ], [ [ 32, 29 ], 4 ], [ [ 32, 30 ], 4 ], [ [ 32, 31 ], 2 ],  [ [ 32, 34 ], 1 ], [ [ 32, 35 ], 1 ], [ [ 32, 39 ], 2 ], [ [ 32, 40 ], 4 ],  [ [ 32, 41 ], 2 ], [ [ 32, 42 ], 8 ], [ [ 32, 43 ], 8 ], [ [ 32, 44 ], 8 ],  [ [ 32, 46 ], 1 ] ]

Under the action of the various automorphism groups
This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives.

gap> G := DirectProduct(DihedralGroup(8),CyclicGroup(2));; gap> A := TrivialGModule(G,GF(2));; gap> A1 := AutomorphismGroup(G);; gap> A2 := GL(1,2);; gap> D := DirectProduct(A1,A2);; gap> P := CompatiblePairs(G,A,D);; gap> M := ExtensionRepresentatives(G,A,P);; gap> List(M,IdGroup); [ [ 32, 46 ], [ 32, 25 ], [ 32, 27 ], [ 32, 28 ], [ 32, 28 ], [ 32, 34 ], [ 32, 40 ], [ 32, 42 ], [ 32, 43 ], [ 32, 44 ], [ 32, 22 ], [ 32, 29 ],  [ 32, 31 ], [ 32, 30 ], [ 32, 39 ], [ 32, 23 ], [ 32, 35 ], [ 32, 41 ] ]