Group Cohomology, Module Extensions, and Group Extensions, and $Ext^2_{\mathbb{Z}G}(\mathbb{Z},A)$

Question

I've read that for some $G$-module $A$, group cohomology can be defined as $$H^{n}(G,A)=Ext^{n}_{\mathbb{Z} G}(\mathbb{Z},A).$$ I've also read that for two $R$-modules $C,D,$ $Ext^{n}_{R}(C,D)$ can be viewed as the equivalence classes of $n$-fold module extensions of $C$ by $D,$ i.e., short exact sequences of modules $$0\to D\to M_1\to M_2\to\cdots\to M_n \to C \to 0.$$ With that in mind, knowing that central extensions of $G$ by some abelian group (viewed as a trivial $G$-module) $A$ are classified by $H^2(G,A)=Ext^2_{\mathbb{Z}G}(\mathbb{Z},A),$ is there a "nice" equivalence between classes of $$0\to A\to M_1\to M_2\to \mathbb{Z}\to 0$$ and central extensions $$1\to A\to E\to G\to 1?$$ Preferably without making recourse to specific resolutions or factor sets.