Let $G$ be a finite group and $S \subseteq G$ a symmetric subset. The Cayley graph $\Gamma(G,S)$ is always vertex-transitive, but it sufficient a simple example to show that it is not always edge-transitive (for example, take $\Gamma(S_3,\{ (12),(123),(132) \})$).
Is it possible to characterize edge-transitive Cayley graphs in terms of some properties of $S$?