On the number of the irreducible characters of factor groups

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. Suppose that ${rm{Irr}} (G | N)$ is the set of the irreducible characters of $G$ that contain $N$ in their kernels. In this paper, we classify solvable groups $G$ in which the set $mathcal{C} (G) = {{rm{Irr}} (G | N) | 1 ne N trianglelefteq G }$ has at most three elements. We also compute the set $mathcal{C}(G)$ for such groups.