A note on the kernels of nonlinear irreducible characters

Let $G$ be a finite nonabelian group, and $\operatorname{Kern}(G)$ be the set of kernels of nonlinear irreducible complex characters of $G$. A nonabelian $p$-group such that all the elements of $\operatorname{Kern}(G)$ have the same order is called a $\mathbb{P}$-group. We give a necessary and sufficient condition for a prime power order group to be a $\mathbb{P}$-group.