  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 $\{ x^{-1}x^{\phi}; \ x\in G\}$. 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 $c,d,e,n$.
