Finite groups in which normality is a transitive relation (Q5943758)

In [Quaest. Math. 19, No. 1-2, 59-82 (1996; Zbl 0859.20017)] \textit{R. W. van der Waall} and \textit{A. Fransman} gave an almost exhaustive treatment of so-called \(T_0\)-groups, i.e., groups in which normality is a transitive relation inside their Frattini factor groups. In this paper the authors give a necessary structure for a finite group \(G\) to be a non-\(T_0\)-group, all of whose proper subgroups are \(T_0\)-groups. Reviewer's remark: It might be possible by looking more deeply, to obtain a structure that is also sufficient for a finite group to be a minimal non-\(T_0\)-group. The reviewer found the proof of Lemma 2.5 too short, in comparison to a convincing proof he has in his possession in preprint form. In short, the interested reader has to work himself/herself through the text in a careful way.