HS-integral and Eisenstein integral normal mixed Cayley graphs

A mixed graph is said to be HS-\emph{integral} if the eigenvalues of its Hermitian-adjacency matrix of the second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set $S$ for which the normal mixed Cayley graph $\text{Cay}(\Gamma, S)$ is HS-integral for any finite group $\Gamma$. We further show that a normal mixed Cayley graph is HS-integral if and only if it is Eisenstein integral. This paper generalizes the results of [M. Kadyan, B. Bhattacharjya. HS-integral and Eisenstein integral mixed Cayley graphs over abelian groups. Linear Algebra Appl. 645:68-90, 2022].