A note on groups paralyzing a subgroup series

We consider groups $\Gamma$ of automorphisms of a group $G$ acting by means of power automorphisms on the factors of a normal series in $G$ with length $m$. We show that $[G, \Gamma]$ is nilpotent with class at most $m$ and that this bound is best possible. Moreover, such a $\Gamma$ is parasoluble with paraheight at most $\frac{1}{2}m(m+3)+1$, provided $\Gamma'$ is periodic. We give best possible bound in the case where the series is a central one.