Zur Struktur endlicher metabelscher Gruppen mit abelscher Fittinggruppe. II (Q791635)

The author defines an \({\mathcal F}\)-group as a finite metabelian group G with an abelian Fitting subgroup F(G). In an earlier paper he described a classification of \({\mathcal F}\)-groups which split over their Fitting subgroups [Part I, ibid. 15, 67-82 (1983; Zbl 0513.20008)], and in this paper he considers the nonsplit case. We can summarize his results as follows. The classification of \({\mathcal F}\)-groups in the nonsplit case can be reduced to the solution of the following problems: (a) Classification of all \({\mathcal F}\)-groups which split over their Fitting subgroups and have trivial centre (this problem was dealt with in the first paper). (b) Determination of all nonsplit extensions K of an abelian group A by some group B of automorphisms of some abelian group F such that (i) A lies in the hypercentre of K, and (ii) the semidirect product of B and F is an \({\mathcal F}\)-group with trivial centre. (c) Determination up to automorphism-equivalence of all homomorphisms from K to B in (b). The author gives an example to show that this approach is practical in some cases, and states that in his dissertation [\textit{G. Steffen}, ''Zur Struktur endlicher metabelscher Gruppen mit abelscher Fittinggruppe'', Diss. Rostock (1978)] he has found all \({\mathcal F}\)-groups whose Fitting subgroup has order at most 50.