In Weibel's book (Exercise 6.9.1), it is claimed that central extensions of the form $0\to A\to X\to G\to 1$ with G perfect are classified by $Hom(H_2(G,\mathbb{Z}), A)$.
My thoughts so far lead me to believe that this group is isomorphic the second cohomology of $G$ with coefficients is $A$ (because $G$ perfect implies $H^1(G,A)= H_1(G,A)= 0$), and so it classifies all extensions of $G$ by $A$ (by the theory of factor sets, see e.g Weibel 6.6)
All that being said I am not seeing how to show that all such extensions are central (which would show what I am trying to prove). My only idea so far has been trying to understand the kernel of the map $X\to G$ using that $G$ is perfect, but I haven't been able to say anything meaningful.
All help is appreciated
Given an extension $$0\to A\to X\to G\to 1$$ $G$ acts on $A$ by conjugation. Now remember that $H^2(G;A)$ classifies all extensions with the given action on $A$. Any extension is central iff $G$ acts trivially on $A$, thus if $A$ has the trivial action, $H^2(G;A)$ classifies central extensions. Note also that Excercise 6.9.1 refers to Exercise 6.1.5, which only talks about $A$ with the trivial $G$-action.