Normal Cayley digraphs on minimal non-abelian groups

We call a Cayley digraph Γ=Cay(G,S) normal for G if GR, the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on non-abelian finite groups G such that every proper subgroup of G is abelian.