I came across the following question which I haven't seen before:
Question. Fix $k\ge 3$. For infinitely many $n$, does there exists a generating set $\langle R_n \rangle = S_n$, $|R_n|=k$, such that the corresponding (undirected) Cayley graph $\Gamma(S_n,R_n)$ is edge-transitive?
Perhaps, there is a simple explicit combinatorial construction which would work for all large enough $n$ and some fixed $k$. Are there any references I am missing?
NOTE: Now that I see a nice answer by Brendan McKay, let me mention the reason for the question. I just learned the construction of edge-transitive expanders on $S_n$ for some sparse sequence of $n$, but my own simple construction to answer the question above worked only for $n=p+1$, where $p$ is a prime.