Let $G$ be a group, and let $A$ be a $G$-module. Then for every subgroup $H$ of $G$, $A$ is also an $H$-module. Furthermore, there's a map $H^2(G,A)\rightarrow H^2(H,A)$.
I would like to know something about those subgroups that satisfy that the image of this map is trivial. In particular I wonder if there is a maximal such $H$.