Abelian groups admitting fixed-point-free automorphisms of order $q^n$

For a wide class of Abelian groups, necessary and sufficient conditions under which a group admits an automorphism of order $q^n$ are found; we also present necessary and sufficient conditions under which a group admits an automorphism $\varphi$ of order $q^n$ such that $\varphi^{qm}$ is a fixed-point-free automorphism for some $m<n$.