Supersolubility of a finite group with normally embedded maximal subgroups in Sylow subgroups

Let $P$ be a subgroup of a Sylow subgroup of a finite group $G$. If $P$ is a Sylow subgroup of some normal subgroup of $G$ then $P$ is called normally embedded in $G$. We establish tests for a finite group $G$ to be $p$-supersoluble provided that every maximal subgroup of a Sylow $p$-subgroup of $X$ is normally embedded in $G$. We study the cases when $X$ is a normal subgroup of $G$, $X=O_{p',p}(H)$, and $X=F^\star(H)$ where $H$ is a normal subgroup of $G$.