A group \(G\) is called a \(\widehat T\)-group in this paper if the set of all subnormal non-normal subgroups of \(G\) satisfies the maximal condition. This notation arises since groups with a transitive normality relation (that is, \(T\)-groups) are obvious examples of \(\widehat T\)-groups. This paper is concerned with the study of \(\widehat T\)-groups. The class of \(T\)-groups has been studied at some length, most notably in papers of W. Gaschütz [J. Reine Angew. Math. 198, 87-92 (1957; Zbl 0077.25003)] and D. J. S. Robinson [Proc. Camb. Philos. Soc. 60, 21-38 (1964; Zbl 0123.24901)], and in the paper under review a number of properties of \(T\)-groups are shown to have an analogue in the class of \(\widehat T\)-groups. For example: finitely generated soluble \(\widehat T\)-groups are polycyclic. As one might expect power automorphisms play an important role. It should also be noted that the authors have also studied the corresponding minimal condition.