Second cohomology group for nontrivial group action of Z4 on Z4

Description of the group
Let $$G$$ be cyclic group:Z4 with generator $$g$$ and $$A$$ be cyclic group:Z4 with generator $$a$$. Note that the automorphism group $$\operatorname{Aut}(A)$$ is isomorphic to cyclic group:Z2, with its non-identity element being the inverse map.

Consider the homomorphism $$\varphi:G \to \operatorname{Aut}(A)$$ that sends $$g$$ to the inverse map of $$A$$. Accordingly, $$\varphi(g^2)$$ is the identity map and $$\varphi(g^3)$$ is also the inverse map of $$A$$.

We are interested in the second cohomology group for the action of $$G$$ on $$A$$, i.e., the group:

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

The cohomology group is isomorphic to cyclic group:Z2.

Elements
We consider here the two cohomology classes, with representative cocycles for each. For simplicity, we choose a representative cocycle that is a normalized 2-cocycle, i.e., if either of the inputs is the identity element of $$G$$, the output is the identity element of $$A$$.