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 $φ$ such that $G=[G,φ]$ and suppos