Weakly 3-DCI abelian groups

Let G be a finite group. A subset S of G not the identity element 1 of G called C I -subset (C( stands for isomorphism"), if for any isomorphism D(G, S) ~ D(G, T), there is f AutG such that T = f(S). G is called a DCI-group if every subset of G {I} is a CI-subset. G is called a GCl-group or simply a C I -group if every inverse-closed subset of G {I} is a CI-subset. (DCI and GCl stands for "digraph Cayley isomorphism" and "Graph Cayley isomorphism" respectively).