a1 Departament D'Algebra, Universitat de Valencia, C/Dr. Moliner 50, 46100 Burjassot (Valencia), Spain
Throughout the paper we consider only finite groups.
J. C. Beidleman and H. Smith  have proposed the following question: “If G is a group and Ha subnormal subgroup of G containing Φ(G), the Frattini subgroup of G, such that H/Φ(G)is supersoluble, is H necessarily supersoluble? “In this paper, we give not only an affirmative answer to this question but also we see that the above result still holds if supersoluble is replaced by any saturated formation containing the class of all nilpotent groups.
(Received November 09 1992)