Involutory automorphisms of finite groups of odd order

For a finite group $G$ we write ${\gamma }_{\mathrm{\infty }}(G)$ to denote the nilpotent residual of $G$, that is, the intersection of all terms of the lower central series. The following theorem is proved: Let $G$ be a finite group of odd order admitting an involutory automorphism $\phi$ such that $G=[G,\phi ]$ and suppose that ${\gamma }_{\mathrm{\infty }}(C_G(\phi ))$ has order $m$. Then the order of ${\gamma }_{\mathrm{\infty }}(G^{\mathrm{\prime }})$ is bounded by a function depending only on $m$. This complements several earlier results on groups of odd order admitting involutory automorphisms.