Cayley graphs for abelian groups have generously transitive automorphism groups. In general a Cayley graph for a non-abelian group will not be generously transitive.
In particular if $G$ is not abelian and $X$ is a so-called graphical regular representation (abbreviated GRR) for $G$, then its automorphism group is not generously transitive. The key property of a GRR is that the stabilizer of a vertex is trivial.
Constructing GRRs is not trivial, but it is expected that most Cayley graphs for groups that are not abelian or generalized dicylic will be GRRs - so choosing a connection set at random usually works.