Let $X$ be a manifold, $G$ and $A$ finite abelian groups and $\epsilon \in H^2(G,A)$ a group cohomology class (for the moment I am assuming there is no action of $G$ on $A$). Given $\alpha \in H^1(X,G)$ there are two natural classes in $H^2(X,A)$ we can construct.

(i) First since $H^2(G,A)$ can be identified with the second cohomology group of $BG$ with $A$ coefficients, and $\alpha$ is associated with a homotopy equivalence class of maps $\widehat{\alpha} :X\rightarrow BG$, we can take the pull-back $\widehat{\alpha}^*\epsilon \in H^2(X,A)$.

(ii) Second, $\epsilon$ identifies uniquely a central extension $$ 1\rightarrow A\rightarrow \widetilde{G}\rightarrow G \rightarrow 1 $$ from which we get the Bockstein map $\beta : H^1(X,G)\rightarrow H^{2}(X,A)$. Then we can construct $\beta(\alpha)\in H^2(X,A)$.

First question: is it true in general that $\widehat{\alpha}^*\epsilon=\beta(\alpha)$? If yes, how to prove it? If no in general, when it happen to be true?

Then I am interested in if and how this story generalizes when $A$ is a non-trivial $G$ module speficied by a homomorphism $\rho :G \rightarrow \text{Aut}(A)$. In this case the twisted group cohomology $H_{\rho}^2(G,A)$ can be identified with the cohomology of $BG$ with local $A$ coefficients $H^2(BG,\widetilde{A})$, and the first construction similarly leads to a class $\widetilde{\alpha}^*\epsilon \in H^2(X,\widetilde{A})$, however I am not sure whether this cohomology group of $X$ with local coefficients makes sense... As for the second construction we again have an extension but this will be non-central. Nevertheless the first Bockstein map $\beta :H^1(X,G)\rightarrow H^2(X,A)$ can still be constructed in the same way and we can consider $\beta(\alpha)$. However it seems to me that this does not live in the cohomology with local $A$ coefficients, but I am really not sure since there might be subtleties in the construction of the Bockstein map. I would be really greatful to anybody could clarify this situation.

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