Normal edge-transitive Cayley graphs and Frattini-like subgroups

For a finite group G and an inverse-closed generating set C of G, let Aut(G;C) consist of those automorphisms of G which leave C invariant. We define an Aut(G;C)-invariant normal subgroup Φ(G;C) of G which has the property that, for any Aut(G;C)-invariant normal set of generators for G, if we remove from it all the elements of Φ(G;C), then the remaining set is still an Aut(G;C)-invariant normal generating set for G. The subgroup Φ(G;C) contains the Frattini subgroup Φ(G) but the inclusion may be proper. The Cayley graph Cay(G,C) is normal edge-transitive if Aut(G;C) acts transitively on the pairs {c,c⁻¹} from C. We show that, for a normal edge-transitive Cayley graph Cay(G,C), its quotient modulo Φ(G;C) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ(G;C) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi.