Groups with supersolvable automorphism group

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.