  Let $G$ be a finite group admitting a coprime automorphism $\alpha$. Let
$J_G(\alpha)$ denote the set of all commutators $[x,\alpha]$, where $x$ belongs
to an $\alpha$-invariant Sylow subgroup of $G$. We show that $[G,\alpha]$ is
soluble or nilpotent if and only if any subgroup generated by a pair of
elements of coprime orders from the set $J_G(\alpha)$ is soluble or nilpotent,
respectively.
