Second cohomology group for trivial group action of Z4 on Z2

Description of the group
We consider here the second cohomology group for trivial group action of specific information about::cyclic group:Z4 on specific information about::cyclic group:Z2, i.e.,

$$\! H^2(G;A)$$

where $$G \cong \mathbb{Z}_4$$ and $$A \cong \mathbb{Z}_2$$.

The cohomology group is isomorphic to cyclic group:Z2.

Computation in terms of group cohomology
The group can be computed as an abstract group by using the group cohomology of cyclic group:Z4 or more generally using group cohomology of finite cyclic groups.

Elements
We list here the elements, grouped by similarity under the action of the automorphism groups on both sides.

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

gap> G := CyclicGroup(4);; 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 := [ [ [ 2, 1 ] ], [ [ 2, 1 ], 0 ] ], orders := [ 2, 2 ], wstack := [ [ 2, 1 ], [ 2, 1 ] ], estack := [ ], pstack := [ 3, 3 ], cstack := [ 1, 1 ], mstack := [ 0, 0 ], list := [ 1, 1 ], module := [ ,  ], mone := , mzero := , avoid := [ ], unavoidable := [ 1, 2, 3 ] ), cohom :=  -> ( GF(2)^1 )>, presentation := rec( group := , relators := [ f1^2*f2^-1, f1^-1*f2*f1*f2^-1, f2^2 ] ) )

Construction of extensions
The extensions can be constructed using the additional command Extensions.

gap> G := CyclicGroup(4);; gap> A := TrivialGModule(G,GF(2));; gap> L := Extensions(G,A);; gap> List(L,IdGroup); [ [ 8, 2 ], [ 8, 1 ] ]

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

gap> G := CyclicGroup(4);; 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); [ [ 8, 2 ], [ 8, 1 ] ]