Normality of 2-Cayley digraphs

A digraph Γ is called a 2-Cayley digraph over a group G, if there exists a semiregular subgroup RG of Aut(Γ) isomorphic to G with two orbits. We say that Γ is normal if RG is a normal subgroup of Aut(Γ). In this paper, we determine the normalizer of RG in Aut(Γ). We show that the automorphism group of each normal 2-Cayley digraph over a group with solvable automorphism group, is solvable. We prove that for each finite group G≠Q8×Z2r, r≥0, where Q8 is the quaternion group of order 8 and Z2 is the cyclic group of order 2, there exists a normal 2-Cayley graph over G and that every finite group has a normal 2-Cayley digraph.