It is well known that given a short exact sequence $1\to H \to G \to G/H \to 1$ the transgression map $$ H^{p-1}(G/H, H^1(H,A)) \to H^{p+1}(G/H,A^H) $$ in the inflation-restriction sequence is in fact the cup product with the opposite of the class of the extension $1 \to H^{ab} \to G/H' \to G/H \to 1$ in $H^2(G/H, H^{ab})$, given that the action of $H$ on the module $A$ is trivial.

What is known about the transgression map in case $A$ is a non-trivial module? Is it possible then to describe transgression in similar terms?