  A finite group $G$ admits an {\em oriented regular representation} if there
exists a Cayley digraph of $G$ such that it has no digons and its automorphism
group is isomorphic to $G$. Let $m$ be a positive integer. In this paper, we
extend the notion of oriented regular representations to oriented
$m$-semiregular representations using $m$-Cayley digraphs. Given a finite group
$G$, an {\em $m$-Cayley digraph} of $G$ is a digraph that has a group of
automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$
orbits. We say that a finite group $G$ admits an {\em oriented $m$-semiregular
representation} if there exists a regular $m$-Cayley digraph of $G$ such that
it has no digons and $G$ is isomorphic to its automorphism group. In this
paper, we classify finite groups admitting an oriented $m$-semiregular
representation for each positive integer $m$.

Group Theory