Centralizers of involutory automorphisms of groups of odd order

The following theorem is proved. Let G be a finite group of odd order admitting an involutory automorphism ϕ such that G=[G,ϕ]. Suppose that CG(ϕ) has a nilpotent subgroup of index n. Then the index [G′:F(G′)] is bounded by a function depending only on n.