Integral Cayley graphs

Let $G$ be a group and $S\subseteq G$ a subset such that $S=S^{-1}$, where $S^{-1}=\{s^{-1}\mid s\in S\}$. Then the Cayley graph $\mathrm{ Cay}(G,S)$ is an undirected graph $\Gamma$ with vertex set $V(\Gamma)=G$ and edge set $E(\Gamma)=\{(g,gs)\mid g\in G, s\in S\}$. For a normal subset $S$ of a finite group $G$ such that $s\in S\Rightarrow s^k\in S$ for every $k\in \mathbb{Z}$ which is coprime to the order of $s$, we prove that all eigenvalues of the adjacency matrix of $\mathrm{ Cay}(G,S)$ are integers. Using this fact, we give affirmative answers to Questions $19.50\mathrm{ (a)}$ and $19.50\mathrm{ (b)}$ in the Kourovka Notebook.