Let $G$ be a non-abelian group which acts trivially on an abelian group $A$. The abelian group $Z^{2}(G,A)$ denotes the set of all $2$-cocycles $c$ on $G$ with values in $A$. Now, let $c\in Z^{2}(G,A)$ and $\alpha\in \mathrm{Hom}(A,G)$. By applying $\alpha$ to the $2$-cocycle condition of $c$, we see that $\alpha\circ c$ satisfies the $2$-cocycle condition. But I'm not sure how we can say that $\alpha\circ c\in Z^{2}(G,G)$. That is, since $G$ is non-abelian, how are $Z^{2}(G,G)$ and $B^{2}(G,G)$ defined?. In which conditions on $G$ such that $\alpha\circ c=1$ for all $c\in Z^{2}(G,A)$ and $\alpha\in \mathrm{Hom}(A,G)$?.
Thank you for any help.