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 Sym(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{Sym}}}(G)$$\end{document}]. 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.