$m$-partite oriented semiregular representation of valency 3 for finite groups

Let $G$ be a finite group and $m\ge 2$ a positive integer. We say that $G$ admits an \emph{oriented $m$-semiregular representation} (abbreviated as OmSR) if there exists a $m$-Cayley digraph $\mathrm{\Gamma }$ over $G$ such that $\mathrm{\Gamma }$ is oriented and $\mathrm{A}\mathrm{u}\mathrm{t}(\mathrm{\Gamma })\cong G$. In \cite{xu1}, we classified finite groups generated by at most two elements that admit an OmSR of valency 3 for $m\ge 2$ and $G≆{\mathbb{Z}}_1$. In this article, we consider $m$-partite digraphs.We say a finite group $G$ admits an \emph{$m$-partite oriented semiregular representation} ($m$-partite digraphical representation), abbreviated as \emph{$m$-POSR} (\emph{$m$-PDR}), if there exists an \emph{oriented} $m$-partite Cayley digraph (\emph{$m$-partite Cayley digraph}) $\mathrm{\Gamma }$ with $\mathrm{A}\mathrm{u}\mathrm{t}(\mathrm{\Gamma })\cong G$. In this paper, we classify finite groups generated by at most two elements that admit $m$-POSR. Since if $G$ admits an $m$-POSR, then $G$ must also admit an $m$-PDR (while the converse does not hold), as a natural consequence, we also provide a complete classification for groups $G=⟨x,y⟩$ that admit $m$-PDR of valency 3. This complements the results in \cite{xu2}.