Babai's problem

Babai's problem is a problem in algebraic [graph theory] first proposed in 1979 by László Babai.

Babai's problem

Let be a finite group, let be the set of all irreducible characters of, let be the Cayley graph corresponding to a generating subset of, and let be a positive integer. Is the set
an invariant of the graph ? In other words, does imply that ?

BI-group

A finite group is called a BI-group if for some inverse closed subsets and of implies that for all positive integers.

Open problem

Which finite groups are BI-groups?