Normal Cayley digraphs on minimal non-abelian groups

We call a Cayley digraph Γ = Cay ( G , S ) normal for G if G R , 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.