Nilpotent length of a finite group admitting a Frobenius group of automorphisms with fixed-point-free kernel

Suppose that a finite group $G$ admits a Frobenius group $FH$ of automorphisms with kernel $F$ and complement $H$ such that the fixed-point subgroup of $F$ is trivial, i.e., $C_G(F)=1$, and the orders of $G$ and $H$ are coprime. It is proved that the nilpotent length of $G$ is equal to the nilpotent length of $C_G(H)$ and the Fitting series of the fixed-point subgroup $C_G(H)$ coincides with a series obtained by taking intersections of $C_G(H)$ with the Fitting series of $G$.