for a finite group $G$ and a trivial abelian $G$ module $A,$ there is the short exact sequence
$0 \rightarrow \mathrm{Ext}^1 (G_{ab},A) \rightarrow H^2(G,A) \rightarrow \mathrm{Hom} (H_2 (G,Z), A) \rightarrow 0$
I'm looking for a description for the connecting maps. Specially, I want to use the representation of $H_2(G,Z)$ as $M(G)=[F,F]\cap R/ [F,R]$ where $G\cong F/R$ and $F$ is free.