A note on groups paralyzing a subgroup series

We consider groups Γ of automorphisms of a groupG acting by means of power automorphisms on the factors of a normal series inG with lengthm. We show that [G, Γ] is nilpotent with class at mostm and that this bound is best possible. Moreover, such a Γ is parasoluble with paraheight at most 1/2m(m+3)+1, provided Γ′ is periodic. We give best possible bound in the case where the series is a central one.