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 Phi(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 Phi(G; C), then the remaining set is still an Aut(G; C)-invariant normal generating set for G. The subgroup Phi(G; C) contains the Frattini subgroup Phi(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 Phi(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 Phi(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. (c) 2021 Elsevier Inc. All rights reserved.