A characterization of finite groups having a single Galois conjugacy class of certain irreducible characters

Let G be a finite group and let Irr(s)(G) be the set of irreducible complex characters ? of G such that ?(1)(2) does not divide the index of the kernel of ?. In this paper, we classify the finite groups G for which any two characters in Irr(s)(G) are Galois conjugate. In particular, we show that such groups are solvable with Fitting height 2.