Characters of finite permutation groups and Krein parameters

Let G be a transitive permutation group on a finite set Ω. If G is multiplicity-free, then EndG(C[Ω]) is commutative, and Krein parameters qi,jk can be defined. Scott proved that if qi,jk≠0, then the corresponding irreducible characters χi,χj,χk of G satisfy (χiχj,χk)≠0. In this paper, we prove the converse of this implication for transitive permutation groups of semidirect product type whose regular normal subgroup is abelian.