  Let $\Gamma$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$,
and let $G=\mathrm{Cay}(\Gamma,T)$ be a Cayley graph of $\Gamma$. The graph $G$
is called normal if $T$ is closed under conjugation. In this paper, we obtain
an upper bound for the second (largest) eigenvalue of the adjacency matrix of
the graph $G$ in terms of the second eigenvalues of certain subgraphs of $G$
(see Theorem 2.6). Using this result, we develop a recursive method to
determine the second eigenvalues of certain Cayley graphs of $S_n$ and we
determine the second eigenvalues of a majority of the connected normal Cayley
graphs (and some of their subgraphs) of $S_n$ with $\max_{\tau\in
T}|\mathrm{supp}(\tau)|\leq 5$, where $\mathrm{supp}(\tau)$ is the set of
points in $[n]$ non-fixed by $\tau$.
