Let $1 \to A \to B \to C \to 1$ be a short exact sequence of (not necessarily abelian) $G$-modules. Passing to non-abelian cohomology, we have the exact sequence of pointed sets $$ 1 \to A^G \to B^G \to C^G \to H^1(G,A) \to H^1(G,B) \to H^1(G,C)$$ My question is
Is there a way to extend this exact sequence a bit further, i.e. up to $H^2$?
I know there is some work that defines $H^2$ in a sensible way, but I'm not whether the above exact sequence can be extended.