Automorphism groups of endomorphism monoids of $0$-unit $3$-valent circulant digraphs

Let $G=\text{Cay}({\mathbb{Z}}_n,S)$ be a $3$-valent circulant digraph with connection set $S=\{0,s,t\}$, where $s,t\in {\mathbb{Z}}_n^{\ast }$ and $t\mathrm{\ne }\pm s$. Determining the automorphism group of its endomorphism monoid reduces to determining the subgroup $U_S({\mathbb{Z}}_n)\le {\mathbb{Z}}_n^{\ast }$ that normalizes $\text{End}(G)$. In this paper, we first establish necessary and sufficient conditions for a unit circulant digraph without $2$-cycles to admit an endomorphism whose image induces a directed cycle. Using this characterization, we explicitly determine $U_S({\mathbb{Z}}_n)$ for the previously unresolved $0$-unit $3$-valent family.