Lattice characterizations of finite supersoluble groups

Let $G$ be a finite group. A subgroup $H$ of $G$ is $\mathfrak {U}$-normal in $G$ if every chief factor of $G$ between $H_{G}$ and $H^{G}$ is cyclic; $H$ is Sylow permutable in $G$ if $H$ commutes with every Sylow subgroup $P$ of $G$, i.e., $HP = PH$. We say that a subgroup $H$ of $G$ is $\mathfrak{U} \wedge sp$-embedded in $G$ if $H = A \cap B$ for some $\mathfrak{U}$-normal subgroup $A$ and Sylow permutable subgroup $B$ in $G$. We find the systems of subgroups $\mathcal L$ in $G$ such that $G$ is supersoluble provided that each $H \in \mathcal L$ is $\mathfrak{U} \wedge sp$-embedded in $G$. In particular, we give new characterizations of finite supersoluble groups.