If $N$ is a normal subgroup of $G$ then there is a coupling: that is, a representation of $G/N$ in $\operatorname{Out}(N)$. In that case, the extensions of $N$ by $G/N$ affording the same coupling are classified by the elements of $H^2(G/N, \,Z(N))$. Given a coupling there may be no extension corresponding: the obstructions are non-zero elements of $H^3(G/N, \, Z(N))$.

Question. Does anyone know a good reference for the functioriality of this theory wrt morphisms of group extensions and in particular to the application to the study of forming Malcev completions of nilpotent normal subgroups in a group?