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 $\mathrm{\Gamma }$ be an abelian group. The $\mathit{\text{mixed}}\mathit{\text{Cayley}}\mathit{\text{graph}}$ $Cay(\mathrm{\Gamma },S)$ is a mixed graph on the vertex set $\mathrm{\Gamma }$ and edge set $\{(a,b):b-a\in S\}$, where $0\in ̸S$. We characterize integral mixed Cayley graph $Cay(\mathrm{\Gamma },S)$ over abelian group in terms of its symbol set $S$.