Consider two groups $G$ and $G'$, where $G$ is the direct product of groups $A$ and $B$, with $B$ abelian, and $G'$ is a nontrivial central extension of $A$ by $B$. Suppose that as groups, $H^1(G,M) \cong H^1(G',M)$ where $M$ is a trivial module in both cases. Furthermore, it can be verified that 1-cocycle representatives in $H^1(G',M)$ are related to 1-cocycle representatives in $H^1(G,M)$ by a bijection $f$- for example, $f$ may correspond to multiplication by an element in $M$.
Given this information, under what conditions can we claim that the cohomology rings $H^*(G,A)$ and $H^*(G',A)$ are isomorphic? I want to use the cup product structure on cohomology, but I don't have a good understanding of what needs to be proved.