Finite \(p\)-groups with few kernels of nonlinear irreducible characters (Q6058274)

Let \(G\) be a finite nonabelian group and \(\mathrm{Kern}(G)\) the set of kernels of nonlinear irreducible characters of \(G.\) Obviously, \(|\mathrm{Kern}(G)|=0\) if and only if \(G\) is abelian while finite groups \(G\) with \(|\mathrm{Kern}(G)|=1\) were determined by \textit{Ya. G. Berkovich} et al. [Characters of finite groups. Part 2. Transl. from the orig. Russian manuscript by Ya. Berkovich and V. Zobina. Transl. edited by David Louvish. Providence, RI: American Mathematical Society (1998; Zbl 0934.20009)]. The authors determine finite \(p\)-groups \(G\) with \(|\mathrm{Kern}(G)|\leq 3.\)