p-Supersolvability and actions on p-groups stabilizing certain subgroups

Let G be a finite group having a noncyclic Sylow p-subgroup of order exceeding pe, where e≥3. If every noncyclic subgroup of G of order pe is normal in G, we show that G is p-supersolvable, and in fact we prove this under the much weaker hypothesis that the noncyclic subgroups of order pe are S-semipermutable in G. The key to the proof is to study the action of a group A on a p-group P under the condition that every noncyclic subgroup of P with order pe is stabilized by A.