Group extensions of $G$ by $A$ $0\to A\to E\to G\to 0$ up to equivalence (where $G$ and $E$ may be nonabelian) are in bijection with the second group cohomology $H^2(G,A)=\text{Ext}^2_{\mathbb{Z}[G]}(\mathbb{Z},A).$
On the other hand, if $G$ is abelian, and we seek only abelian extensions, such extensions are (up to equivalence) in bijection with $\text{Ext}^1_{\mathbb{Z}}(G,A)$ (whence the $\text{Ext}$ functor gets its name).
Hence there is an inclusion $\text{Ext}^1_{\mathbb{Z}}(G,A)\hookrightarrow \text{Ext}^2_{\mathbb{Z}[G]}(\mathbb{Z},A).$ Perhaps we could say it’s induced by the inclusion functor of abelian groups into groups, acting on group extensions. Other than the description given, using the bijection to group extensions, is there another way to see this map?