Non-normal edge-transitive directed Cayley graphs of abelian groups

For a group G, and a subset S of G such that 1G 62 S, let X = Cay(G; S) be the corresponding Cayley graph. Then X is said to be normal edge transitive if NAut(X)(G) is transitive on edges. In this paper, we determine all connected directed Cayley graphs of finite abelian groups with valency at most 3 which are normal edge transitive but not normal.