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.