Finite groups with coprime fixed-point-free automorphisms and applications

We study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\text{Aut}(G)$. In this paper, we show that such groups are solvable. As an application, we characterise a class of normal edge-transitive Cayley graphs of $G$, which are complete multipartite graphs.