Some results on 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 $\mathrm{\mid }cd(G)\mathrm{\mid }=2$, where $cd(G)=\{\chi (1)\mathrm{\mid }\chi \in Irr(G)\}$. We call such a group a GVZ-group with two character degrees. We establish bijections between the sets of characters of some groups obtained from a GVZ-group with two character degrees. Additionally we obtain some alternate characterizations of a GVZ-group with two character degrees and we construct a GVZ-group having the character degree set $\{1,p\}$.