Criteria for solubility and nilpotency of finite groups with automorphisms

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.