Second cohomology group for nontrivial group action of Z2 on Z4

Description of the group
Let $$G$$ be cyclic group:Z2 and $$A$$ be cyclic group:Z4. Note that $$\operatorname{Aut} A$$ is isomorphic to the group cyclic group:Z2, and it comprises the identity automorphism and the automorphism sending every element of $$A$$ to its inverse. Consider the homomorphism $$\varphi:G \to \operatorname{Aut}(A)$$ that is an isomorphism: it sends the non-identity element of $$G$$ to the inverse map on $$A$$, and the identity element of $$G$$ to the identity automorphism of $$A$$.

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

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

The cohomology group is isomorphic to cyclic group:Z2.

Elements
Let $$g$$ denote the non-identity element of $$G$$ and $$a$$ denote a generator for $$A$$. 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 $$A$$. Thus, to specify the cocycle $$f$$, we need only specify $$f(g,g)$$.

Group actions
Because each of the cohomology class types has size one, thereis no scope for permutation of these under any group actions.