Normal Edge-Transitive Cayley Graphs of Frobenius Groups

A Cayley graph for a group G is called normal edge-transitive if it admits an edge-transitive action of some subgroup of the holomorph of G [the normaliser of a regular copy of G in $${{\mathrm{Sym}}}(G)$$Sym(G)]. We complete the classification of normal edge-transitive Cayley graphs of order a product of two primes by dealing with Cayley graphs for Frobenius groups of such orders. We determine the automorphism groups of these graphs, proving in particular that there is a unique vertex-primitive example, namely the flag graph of the Fano plane.