Let $G$ be a group and $A$ a $G$-module. It well know that there is a group isomorphism between the second cohomologoy group $H^2(G,A)$ and the abelian group $OpExt(G,A)$ of classes of extension $$1\to A\to E\to G\to 1$$ of $G$ by the abelian group $A$.

Define over $H^2(G,A)$ the equivalence relation where two cohomology classes are equivalent if for the associated extensions $1\to A\to E_1\to G\to 1$, $1\to A\to E_2\to G\to 1$, $E_1\cong E_2$ as groups.

Question:There is a systematic way in order to decide if two elements in $H^2(G,A)$ are equivalent?

such that the induced action of $G$ onto $A$ is the one you started with. Ot $\endgroup$