Supersolvable Frobenius groups with nilpotent centralizers

Let $FH$ be a supersolvable Frobenius group with kernel $F$ and complement $H$. Suppose that a finite group $G$ admits $FH$ as a group of automorphisms in such a manner that $C_G(F)=1$ and $C_G(H)$ is nilpotent of class $c$. We show that $G$ is nilpotent of $(c,\mid FH\mid )$-bounded class.