  Assume that a finite group $G$ admits a Frobenius group of automorphisms $FH$
with kernel $F$ and complement $H$ such that $C_{G}(F)=1$. In this paper, we
investigate this situation and prove that if $C_G(H)$ is supersoluble and
$C_{G'}(H)$ is nilpotent, then $G$ is supersoluble. Also, we show that $G$ is a
Sylow tower group of a certain type if $C_{G}(H)$ is a Sylow tower group of the
same type.
