Exponent of a finite group admitting a coprime automorphism

Let $G$ be a finite group admitting a coprime automorphism $phi$ of order $n$. Denote by $G_{phi}$ the centralizer of $phi$ in $G$ and by $G_{-phi}$ the set ${ x^{-1}x^{phi}; xin G}$. We prove the following results. 1. If every element from $G_{phi}cup G_{-phi}$ is contained in a $phi$-invariant subgroup of exponent dividing $e$, then the exponent of $G$ is $(e,n)$-bounded. 2. Suppose that $G_{phi}$ is nilpotent of class $c$. If $x^{e}=1$ for each $x in G_{-phi}$ and any two elements of $G_{-phi}$ are contained in a $phi$-invariant soluble subgroup of derived length $d$, then the exponent of $[G,phi]$ is bounded in terms of $c,d,e,n$.