If $G$ is a finite $p$-group and $H$ is a proper subgroup of $G$, then, it is well known that the union of conjugates of $H$ in $G$ is proper subset of $G$. The problem I considering is the following:

**Question** Let $G$ be a finite $p$-group, $H$ be a non-normal subgroup, such that its normalizer $N_G(H)$ is normal in $G$ (hence, all conjugates of $H$ are contained in $N_G(H)$). Is the union of all conjugates of $H$, a proper subset of $N_G(H)$?