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Let  be a finite group and letIrrs⁢(G)\mathrm{Irr}_{\mathfrak{s}}(G)be the set of irreducible complex characters  of  such thatχ⁢(1)2\chi(1)^{2}does not divide the index of the kernel of . In this paper, we classify the finite groups  for which any two characters inIrrs⁢(G)\mathrm{Irr}_{\mathfrak{s}}(G)are Galois conjugate. In particular, we show that such groups are solvable with Fitting height 2.

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