Non-abelian finite groups whose character sums are invariant but are not Cayley isomorphism

Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. By a Cayley graph $Cay(G,S)$ we mean the graph whose vertex set is the set of elements of $G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. A group $G$ is called a CI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $S^{\alpha }=T$ for some automorphism $\alpha$ of $G$. A finite group $G$ is called a BI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $M_{\nu }^S=M_{\nu }^T$ for all positive integers $\nu$, where $M_{\nu }^S$ denotes the set $\big{\sum_{s\in S}\chi(s) | \chi(1)=\nu, \chi $\text{ is a complex irreducible character of }$ G \big}$. It was asked by L'aszl'o Babai [$\mathit{\text{J.}}\mathit{\text{Combin.}}\mathit{\text{Theory}}\mathit{\text{Ser.}}\mathit{\text{B}}$, {\bf 27} (1979) 180-189] if every finite group is a BI-group; various examples of finite non BI-groups are presented in [$\mathit{\text{Comm.}}\mathit{\text{Algebra}}$, {\bf 43} (12) (2015) 5159-5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However it is known that there are finite abelian non CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders $20$ and $42$ which are not CI-groups. We also list all BI-groups of orders up to $30$.