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 \{1}. By a Cayleygraph 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−1y ∈ S. A group G is called a CI-group if Cay(G,S) ∼ = Cay(G,T) for some inverse closed subsets S and T of G\{1}, then Sα = T for some automorphism α of G. A finite group G is called a BI-group if Cay(G,S) ∼ = Cay(G,T) for some inverse closed subsets S and T of G \{1}, then MS ν = MT ν for all positive integers ν, where MS ν denotes the set {∑s∈S χ(s)|χ(1) = ν,χ is a complex irreducible character of G}. It was asked by László Babai [J. Combin. Theory Ser. B, 27 (1979) 180-189] if every finite group is a BI-group; various examples of finite non BI-groups are presented in [Comm. Algebra, 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 knownthattherearefiniteabeliannonCI-groups. Existenceofafinitenon-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.