A characterization of finite groups having a single Galois conjugacy class of certain irreducible characters (Q6146263)

Let \(G\) be a finite group. Let \(\mathrm{Irr}(G)\) be the set of all irreducible complex characters of \(G\) and let \[\mathrm{Irr}_{\mathfrak{n}}(G)=\{ \chi \in \mathrm{Irr}(G) \mid |G:\mathrm{ker}(\chi)|\cdot \chi(1)^{-2} \in \mathbb{N} \}\] and \[\mathrm{Irr}_{\mathfrak{s}}(G)=\mathrm{Irr}(G)\setminus \mathrm{Irr}_{\mathfrak{n}}(G).\] \textit{S. M. Gagola} and \textit{M. L. Lewis} in [Commun. Algebra 27, No. 3, 1053--1056 (1999; Zbl 0929.20010)] proved that a finite group \(G\) is nilpotent if and only if \(|\mathrm{Irr}_{\mathfrak{s}}(G)|=0\). Subsequently \textit{H. Lv} et al. [J. Group Theory 25, No. 4, 727--740 (2022; Zbl 1506.20004)] classified the finite groups \(G\) with \(|\mathrm{Irr}_{\mathfrak{s}}(G)|=1\), which turn out to be solvable groups with Fitting height \(2\). In the paper under review, the authors classify the finite groups \(G\) for which any two characters in \(\mathrm{Irr}_{\mathfrak{s}}(G)\) are Galois conjugate. In particular, they show that such groups are solvable with Fitting height \(2\).