# For a $G$-module $A$ is there a maximal subgroup $H$ of $G$ such that the image $H^2(G,A)\rightarrow H^2(H,A)=0$?

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 $H$.

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If your group finite? –  Mariano Suárez-Alvarez Jan 23 '12 at 3:37
No, not necessarily. Although in the cases that I'm thinking of A is finite. –  Nicole Jan 23 '12 at 3:38
Just out of curiosity, what would you have said under the assumption that $G$ is finite? –  Nicole Jan 23 '12 at 5:22
For the finite case, there is no unique maximal such $H$; just consider $A_5$ with trivial module over $\mathbb{F}_2$, and think about $H$ being a Sylow 3-group or 5-group. –  user641 Jan 23 '12 at 5:30
Yeah, that was where I was going :) In the infinite case: I think there are non-free groups such that every proper subgroup is free. –  Mariano Suárez-Alvarez Jan 23 '12 at 6:58