Groups with supersolvable automorphism group

Abstract

We call a finite group G ultrasolvable if it has a characteristic subgroup series whose factors are cyclic. It was shown by Durbin--McDonald that the automorphism group of an ultrasolvable group is supersolvable. The converse statement was established by Baartmans--Woeppel under the hypothesis that G has no direct factor isomorphic to the Klein four-group. We extend this result by proving that Aut(G) is supersolvable if and only if G is ultrasolvable or G=H\times C_2\times C_2 where H is ultrasolvable of odd order. This corrects an erroneous claim by Corsi Tani. Our proof is more elementary than Baartmans--Woeppel's and uses some ideas of Corsi Tani and Laue.