Existence of DCI-groups that are not CI(2)-groups

Determine whether there exists a DCI-group that is not a CI(2)-group.

Background

A finite group is a DCI-group if every Cayley digraph over it has the Cayley isomorphism property, whereas a CI(2)-group is defined through the stronger transjugacy condition for every 2-closed permutation group containing the regular representation. The paper notes that every CI(2)-group is DCI, but the converse is not known. Resolving this question would clarify whether the 2-closure framework captures all DCI-groups.

References

Clearly, every CI(2)-group is also DCI, however it is not known whether there exists a DCI-group which is non-CI(2).

— On CI-property of normal Cayley digraphs over abelian groups  (2503.00859 - Ryabov, 2 Mar 2025) in Page 3, Introduction, Section 1