Group extensions with non-abelian kernel

Question

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?

Answer 1

There is an account of the general theory of group extensions with non-abelian kernel in Gruenberg's Springer Lecture Notes 143, Cohomological Topics in Group Theory: here he refers to his own paper 'A New Treatment of Group Extensions, Math. Zeit. 102, 1967, 340--350. It's possible that Gruenberg was a pioneer in this field as well as taking the novel approach using what we now call the 'Gruenberg resolution'. I have not explored Schreier's paper from 1926: it would be surprising if this included more than a fragment of what Gruenberg knew but I am ready to be corrected. Eilenberg and Mac Lane seem to confine themselves largely to the case of abelian kernel. More modern treatments include D.J.S.Robinson's in 'A Course in the Theory of Groups' Graduate Texts in Mathematics 80, Spriner, and Robinson's treatment follows that of Gruenberg moderately closely. The idea of applying this theory to the behaviour of groups under Mal'cev completion of their Fitting subgroups must have been widely imagined and it would be interesting to know if there is any account of this specific direction.