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 $G_{ph
i}cup G_{-phi}$ 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 $c,d,e,n$.
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.
