Let $G$ be an abelian group, $A$ a trivial $G$-module. We know that $\text{Ext}(G,A)$ classifies abelian extensions of $G$ by $A$, whereas $H^2(G,A)$ classifies central extensions of $G$ by $A$. So we have a canonical inclusion $\text{Ext}(G,A)\hookrightarrow H^2(G,A)$. Is there some naturally arising exact sequence/spectral sequence which realizes this injection?


Usually this kind of thing can be explained by constructing a clever short exact sequence, but here I have no idea how one might compare $R^1\text{Hom}\_\mathbb{Z}(G,\underline{\quad})$ with $R^2\text{Hom}\_G(\mathbb{Z},\underline{\quad})$.



Injection of Ext into H^2
You get a description from the universal coefficient theorem which gives a (split) exact sequence
$$
0\to \mathrm{Ext}(H\_1(G),A) \to H^2(G,A) \to \mathrm{Hom}(H\_2(G),A) \to 0
$$
and the fact that $H\_1(G)=G$. We have that $H\_2(G)=\Lambda^2G$ and the map $H^2(G,A) \to \mathrm{Hom}(H\_2(G),A)$ associates to an extension its commutator map.


