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 $phi$. We obtain two results bounding the exponent of $[G,phi]$. Denote by $G_{-phi}$ the set ${[g,phi],vert, gin G}$ and by $G_{phi}$ the centralizer of $phi$, that is, the subgroup of fixed points of $phi$. The obtained results are as follows.1. Assume that the subgroup $langle x,yrangle$ has derived length at most $d$ and $x^e=1$ for every $x,yin G_{-phi}$. Suppose that $G_phi$ is nilpotent of class $c$. Then the exponent of $[G,phi]$ is $(c,d,e)$-bounded.2. Assume that $G_phi$ has rank $r$ and $x^e=1$ for each $xin G_{-phi}$. Then the exponent of $[G,phi]$ is $(e,r)$-bounded.