Consider a short exact sequence of (finite, or pro-finite) groups $ 1 \to H \to G \xrightarrow{\pi} G/H \to 1 $ and suppose for simplicity that $H$ lies in the centre of $G$. Such short exact sequence corresponds to a 2-cocycle in $H^2(G/H,H)$ under the correspodence between central extensions and second group cohomology. The cocycle is calculated as follows: take a section $j$ of $\pi$ and let $ h_{\sigma,\tau}=j(\sigma)+\sigma j(\tau) - j(\sigma\tau) $
My question is about a map $ f: H^1(G,A^H) \to H^2(G/H,A^H) $ which is defined by a somewhat similar formula.
Take $\{g_\sigma\} \in H^1(G,A^H)$, take a section $j$ as above, and define the 2-cocycle as follows $ h_{\sigma,\tau}=g_{j(\sigma)}+j(\sigma)g_{j(\tau)}-g_{j(\sigma\tau)} $ One can check that the class of $h$ does not depend neither on a particular choice of a cocycle $g$ in a cohomology class, nor on the section $j$, the map is also functorial in $A$.
What is known about this map? Is it a part of some standard exact equence, or a composition of some well-known functors (like restriction/corestriction) ?