Fitting height of a finite group with a Frobenius group of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms FH with kernel F and complement H such that F acts without nontrivial fixed points (that is, such that CG(F)=1). It is proved that the Fitting height of G is equal to the Fitting height of the fixed-point subgroup CG(H) and the Fitting series of CG(H) coincides with the intersections of CG(H) with the Fitting series of G. As a corollary, it is also proved that for any set of primes π the π-length of G is equal to the π-length of CG(H).