  Using Frobenius normal forms of matrices over finite fields as well as the
Burnside Basis Theorem, we give a direct proof of Horo\v{s}evski\u{i}'s result
that every automorphism $\alpha$ of a finite nilpotent group has a cycle whose
length coincides with $\mathrm{ord}(\alpha)$. Also, we give two new sufficient
conditions for an automorphism $\alpha$ of an arbitrary finite group to satisfy
this property, namely when $\mathrm{ord}(\alpha)$ is a product of at most two
prime powers or when $\alpha$ has a sufficiently large cycle. This will allow
us to show that the least order of a group where this property is violated is
120. Finally, we observe that any finite group embeds both into a group with
this property (as all finite symmetric groups enjoy the property) as well as
into a finite group not having this property.
