Finite groups with few character values that are not character degrees

Let $G$ be a finite group and $\chi \in \mathrm{Irr}(G)$. Define $\mathrm{cv}(G)=\{\chi(g)\mid \chi \in \mathrm{Irr}(G), g\in G \}$, $\mathrm{cv}(\chi)=\{\chi(g)\mid g\in G \}$ and denote $\mathrm{dl}(G)$ by the derived length of $G$. In the 1990s Berkovich, Chillag and Zhmud described groups $G$ in which $|\mathrm{cv}(\chi)|=3$ for every non-linear $\chi \in \mathrm{Irr}(G)$ and their results show that $G$ is solvable. They also considered groups in which $|\mathrm{cv}(\chi)|=4$ for some non-linear $\chi \in \mathrm{Irr}(G)$. Continuing with their work, in this article, we prove that if $|\mathrm{cv}(\chi)|\leqslant 4$ for every non-linear $\chi \in \mathrm{Irr}(G)$, then $G$ is solvable. We also considered groups $G$ such that $|\mathrm{cv}(G)\setminus \mathrm{cd}(G)|=2$. T. Sakurai classified these groups in the case when $|\mathrm{cd}(G)|=2$. We show that $G$ is solvable and we classify groups $G$ when $|\mathrm{cd}(G)|\leqslant 4$ or $\mathrm{dl}(G)\leqslant 3$. It is interesting to note that these groups are such that $|\mathrm{cv}(\chi)|\leqslant 4$ for all $\chi \in \mathrm{Irr}(G)$. Lastly, we consider finite groups $G$ with $|\mathrm{cv}(G)\setminus \mathrm{cd}(G)|=3$. For nilpotent groups, we obtain a characterization which is also connected to the work of Berkovich, Chillag and Zhmud. For non-nilpotent groups, we obtain the structure of $G$ when $\mathrm{dl}(G)=2$.