Involutory automorphisms of finite groups of odd order

For a finite group $G$ we write $γ_\infty(G)$ 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 $\varphi$ such that $G=[G,\varphi]$ and suppose that $γ_\infty(C_G(\varphi))$ has order $m$. Then the order of $γ_\infty(G')$ is bounded by a function depending only on $m$. This complements several earlier results on groups of odd order admitting involutory automorphisms.