Second cohomology group for nontrivial group action of Z2 on V4

Description of the group
Let $$G$$ be cyclic group:Z2 and $$A$$ be the Klein four-group. Consider the homomorphism of groups $$\varphi:G \to \operatorname{Aut}(A)$$ defined as follows: the homomorphism sends the non-identity element of $$G$$ to an automorphism that interchanges two of the elements of order two in $$A$$ (if we think of $$A$$ as an internal direct product of two copies of cyclic group:Z2, this can be viewed as a coordinate exchange automorphism). Another way of thinking of this is that, knowing that $$\operatorname{Aut}(A) \cong S_3$$ (the symmetric group:S3), the homomorphism is an injective one with image one of the copies of S2 in S3.

We are interested in the second cohomology group for the action $$\varphi$$, i.e., the group:

$$H^2_\varphi(G,A)$$

The group is isomorphic to the trivial group.

Elements
The group is the trivial group and has only one element: its identity element. The group extension corresponding to this element is dihedral group:D8 (ID: (8,3)). We can choose the zero cocycle as a representative cocycle, but there is another candidate for a representative normalized 2-cocycle: a candidate $$f$$ such that $$f(g,g)$$ is the fixed element of $$A$$ where $$g$$ is the non-identity element of $$G$$. The key thing to note is that this still represents the zero cohomology class because it is a 2-coboundary.