Exponent of a finite group of odd order with an involutory automorphism

Let $G$ be a finite group of odd order admitting an involutory automorphism $\varphi$. We obtain two results bounding the exponent of $[G,\varphi ]$. Denote by $G_{-\varphi }$ the set $\{[g,\varphi ]\mathrm{\mid }g\in G\}$ and by $G_{\varphi }$ the centralizer of $\varphi$, that is, the subgroup of fixed points of $\varphi$. The obtained results are as follows.1. Assume that the subgroup $\langle x,y\rangle$hasderivedlengthatmost$d$and$x^e=1$forevery$x,y\in G_{-\phi}$\mathrm{.}Supposethat$G_\phi$isnilpotentofclass$c$. Then the exponent of $[G,\varphi ]$ is $(c,d,e)$-bounded.2. Assume that $G_{\varphi }$ has rank $r$ and $x^e=1$ for each $x\in G_{-\varphi }$. Then the exponent of $[G,\varphi ]$ is $(e,r)$-bounded.