Decomposable Groups with Few Nonlinear Irreducible Character Kernels

Let \({{\,\textrm{Kern}\,}}(G)\) $\text{Kern}(G)$ denote the set of nonlinear irreducible character kernels of a finite group G. In this paper, we classify decomposable groups G with \(|{{\,\textrm{Kern}\,}}(G)|\le 3\) $|\text{Kern}(G)|\le 3$ . In particular, nilpotent groups G with \(|{{\,\textrm{Kern}\,}}(G)|\le 3\) $|\text{Kern}(G)|\le 3$ are determined.