Let $G$ be a group with normal subgroup $H$ and let $M$ be an abelian group on which $G$ acts. The associated inflation-restriction exact sequence is $$1 \to H^1(G/H, M^H) \to H^1(G, M) \to H^1(H, M)^{G/H} \to H^2(G/H, M^H).$$ This can in fact be extended to an explicit seven-term exact sequence; see this paper.
Now let us drop the assumption that $M$ is abelian. One can still define cohomology functors $H^i$. For $H^0$, $H^1$, and $H^2$ this is done for example in Giraud's "Cohomologie non abélienne".
To what extent does there exist an inflation-restriction exact sequence in the non-abelian setting?