 Nonabelian first cohomology

For a group $G$ acting by automorphisms on a possibly noncommutative group $B$, $H^1(G,B)$ is the pointed set of maps satisfying $c_{\sigma \tau }=c_{\sigma }\sigma (c_{\tau })$ modulo $c_{\sigma }\mapsto b{c}_{\sigma }\sigma (b{)}^{-1}$. It classifies twisting of a reference action. It need not be an abelian group.