Coprime automorphisms of finite groups

Let $G$ be a finite group admitting a coprime automorphism $\alpha$ of order $e$. Denote by $I_G(\alpha)$ the set of commutators $g^{-1}g^\alpha$, where $g\in G$, and by $[G,\alpha]$ the subgroup generated by $I_G(\alpha)$. We study the impact of $I_G(\alpha)$ on the structure of $[G,\alpha]$. Suppose that each subgroup generated by a subset of $I_G(\alpha)$ can be generated by at most $r$ elements. We show that the rank of $[G,\alpha]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(\alpha)$ has odd order, then $[G,\alpha]$ has odd order too. Further, if every pair of elements from $I_G(\alpha)$ generates a soluble, or nilpotent, subgroup, then $[G,\alpha]$ is soluble, or respectively nilpotent.