Integral mixed cayley graph over abelian group

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The \textit{mixed Cayley graph} $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and edge set $\{ (a,b): b-a\in S \}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(\Gamma,S)$ over abelian group in terms of its symbol set $S$.