Babai's problem

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

Let ${\displaystyle G}$be a finite group, let ${\displaystyle \operatorname {Irr} (G)}$be the set of all irreducible characters of ${\displaystyle G}$, let ${\displaystyle \Gamma =\operatorname {Cay} (G,S)}$be the Cayley graph (or directed Cayley graph) corresponding to a generating subset ${\displaystyle S}$of ${\displaystyle G\setminus \{1\}}$, and let ${\displaystyle \nu }$be a positive integer. Is the set

${\displaystyle M_{\nu }^{S}=\left\{\sum _{s\in S}\chi (s)\;|\;\chi \in \operatorname {Irr} (G),\;\chi (1)=\nu \right\}}$

an invariant of the graph ${\displaystyle \Gamma }$? In other words, does ${\displaystyle \operatorname {Cay} (G,S)\cong \operatorname {Cay} (G,S')}$imply that ${\displaystyle M_{\nu }^{S}=M_{\nu }^{S'}}$?

A finite group ${\displaystyle G}$is called a BI-group (Babai Invariant group) if ${\displaystyle \operatorname {Cay} (G,S)\cong \operatorname {Cay} (G,T)}$for some inverse closed subsets ${\displaystyle S}$and ${\displaystyle T}$of ${\displaystyle G\setminus \{1\}}$implies that ${\displaystyle M_{\nu }^{S}=M_{\nu }^{T}}$for all positive integers ${\displaystyle \nu }$.

Which finite groups are BI-groups?

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