  Given a group $G$, an {\em $m$-Cayley digraph $\G$ over $G$} is a digraph
that has a group of automorphisms isomorphic to $G$ acting semiregularly on the
vertex set with $m$ orbits. We say that $G$ admits an {\em oriented
$m$-semiregular representation} (O$m$SR for short), if there exists a regular
$m$-Cayley digraph $\G$ over $G$ such that $\G$ is oriented and its
automorphism group is isomorphic to $G$. In particular, O$1$SR is also named as
ORR. Verret and Xia gave a classification of finite simple groups admitting an
ORR of valency two in [Ars Math. Contemp. 22 (2022), \#P1.07]. Let $m\geq 2$ be
an integer. In this paper, we show that all finite groups generated by at most
two elements admit an O$m$SR of valency two except four groups of small orders.
Consequently, a classification of finite simple groups admitting an O$m$SR of
valency two is obtained.
