When is the restriction map in cohomology an isomorphism?

When is the restriction map in cohomology an isomorphism? Let $$G$$ be a group, $$H$$ one of its normal subgroups and $$M$$ a $$G$$-module. If $$H^1(G,M)\simeq H^1(H,M)^{G/H}$$, can we conclude that $$H=G$$?

• I wouldn't advise it. – Ryan Budney Mar 14 at 15:29
• mathoverflow.net/questions/95036/… – Venkataramana Mar 14 at 15:36
• see the above link; there is an exact sequence which tells you that in general, there is a map from $H^1(G,M)$ into $H^1(H,M)^{G/H}$ with kernel $H^1(G/H,M^H)$. – Venkataramana Mar 14 at 15:38