On the number of Heisenberg characters of finite groups

An irreducible character $\chi$ of a finite group $G$ is called a Heisenberg character if $\ker \chi \supseteq [G, [G, G]]$. In this paper, we prove that the group $G$ has exactly $r$, $r \leq 3$, Heisenberg characters if and only if $|{G}/{G'}|=r$. If $G$ has exactly four Heisenberg characters, then $|{G}/{G'}|=4$, but the converse is not correct in general. Finally, it is proved that if $G$ has exactly five Heisenberg characters, then $|{G}/{G'}|=5$ or $|{G}/{G'}|=4$ and one of the Heisenberg characters of $G$ has the degree $2$.