Finite groups with $\mathbb{P}$-subnormal biprimary subgroups

In this paper we study finite groups with $\mathbb{P}$-subnormal biprimary dispersive subgroups. We prove that a group all of whose biprimary $p$-closed $pd$-subgroups are $\mathbb{P}$-subnormal is $p$-solvable, where $p$ is the largest prime divisor of the order of the group. We also prove that a group with biprimary $2$-nilpotent $\mathbb{P}$-subnormal $2d$-subgroups is solvable.