On $p$-groups with one nonlinear codegree

For an irreducible character $\chi$ of a finite group $G$, its codegree is defined by $cod(\chi )=\mathrm{\mid }G:\ker \chi \mathrm{\mid }\mathrm{/}\chi (1)$. In this paper, we characterize fnite $p$-groups whose nonlinear irreducible characters all have the same codegree.