Let $G$ be a finite group, and $K \lhd G$. I want to prove that there is a bijection between the set of characters of $G$ that correspond to representations $\varrho$ of $G$ with $K \leqslant \ker \varrho$ and the set of irreducible characters of $G/K$.
For such a representation $\varrho$ of $G$, it is easy to show that $\tilde{\varrho} : gK \mapsto \varrho(g)$ is well-defined and a representation of $G/K$. Conversely, given a representation $\pi$ of $G/K$, the map $g \mapsto \pi(gK)$ is clearly a representation of $G$ with $K$ a subgroup of its kernel, and these constructions are inverse to one another.
Let $L$ denote the set of characters of $G$ that correspond to representations $\varrho$ of $G$ with $K \leqslant \ker \varrho$, and let $M$ be the set of characters of $G/K$. Then, it is easy to show that the mapping that sends $\chi \in L$ for $\chi$ the character of $\varrho$ to the character of $\tilde{\varrho}$ is well-defined and injective. By the inverse construction given above, every element in $M$ has a preimage in $L$.
This is where I am stuck: It seems that there is a bijection between $L$ and all characters of $G/K$, not just the irreducible ones. There must be something wrong about my reasoning, but I am not sure where.