Finite Edge-Transitive Cayley Graphs, Quotient Graphs and FrattiniGroups

The edge-transitivity of a cayley graph is most easily recognisable if the subgroup of affine mapspreserving the graph structure is itself edge-transitive. these are the so-called normal edge-transitivecayley graphs. each of them determines a set of quotients which are themselves normal edge-transitivecayley graphs and, and which are built from a very restricted family of groups (direct products of simplegroups). we address the questions: how much information about the original cayley graph can we retrievefrom this special set of quotients? and can we ever reconstruct the original cayley graph from them: ifso, then how?our answers to these questions involve a type of relative frattini subgroup determined by the cayleygraph, which has similar properties to the frattini subgroup of a finite group ill discuss this and givesome examples. it raises many new questions about cayley graphs.