GVZ-groups with two character degrees

We investigate the finite groups $G$ for which $\chi (1{)}^2=\mathrm{\mid }G:Z(\chi )\mathrm{\mid }$ for all characters $\chi \in Irr(G)$ and $\mathrm{\mid }cd(G)\mathrm{\mid }=2$. We obtain some alternate characterizations of these groups and we obtain some information regarding the structure of these groups.