Finite groups with some subgroups of Sylow subgroups weakly $\mathcal{H}$-embedded

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is an $\mathcal{H}$-subgroup in $G$ if $N_{G}(H)\cap H^{g}\leq H$, for all $g\in G$. The subgroup $H$ is called weakly $\mathcal{H}$-embedded in $G$ if $G$ has a normal subgroup $K$ such that $H^{G}=HK$ and $H\cap K$ is an $\mathcal{H}$-subgroup in $G$, where $H^{G}$ is the normal closure of $H$ in $G$, that is, $H^{G}=<H^{g}:g\in G>$. Using this concept, we improve and extend Theorem 1.6 and Corollary 1.9 of [3] and Theorem 3.1 of [17].