  Following the literature, a group $G$ is called a group of central type if
$G$ has an irreducible character that vanishes on $G\setminus Z(G)$. Motivated
by this definition, we say that a character $\chi\in {\rm Irr}(G)$ has central
type if $\chi$ vanishes on $G\setminus Z(\chi)$, where $Z(\chi)$ is the center
of $\chi$. Groups where every irreducible character has central type have been
studied previously under the name GVZ-groups (and several other names) in the
literature. In this paper, we study the groups $G$ that possess a nontrivial,
normal subgroup $N$ such that every character of $G$ either contains $N$ in its
kernel or has central type. The structure of these groups is surprisingly
limited and has many aspects in common with both central type groups and
GVZ-groups.
