Babai's problem

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

Babai's problem
Let $$G$$ be a finite group, let $$\operatorname{Irr}(G)$$ be the set of all irreducible characters of $$G$$, let $$\Gamma=\operatorname{Cay}(G,S)$$ be the Cayley graph (or directed Cayley graph) corresponding to a generating subset $$S$$ of $$G\setminus \{1\}$$, and let $$\nu$$ be a positive integer. Is the set
 * $$M_\nu^S=\left\{\sum_{s\in S} \chi(s)\;|\; \chi\in \operatorname{Irr}(G),\; \chi(1)=\nu \right\}$$

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

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

Open problem
Which finite groups are BI-groups?