Finite groups with supersoluble subgroups of given orders

We study a finite group $G$ with the following property: for any of its maximal subgroups $H$, there exists a subgroup $H_1$ such that $|H_1|=|H|$ and $H_1\in \frak F$, where $\frak F$ is the formation of all nilpotent groups or all supersoluble groups. We prove that, if $\frak F=\frak N$ is the formation of all nilpotent groups and $G$ is nonnilpotent, then $|\pi (G)|=2$ and $G$ has a normal Sylow subgroup. For the formation $\frak F=\frak U$ of all supersoluble groups and a soluble group $G$ with the above property, we prove that $G$ is supersoluble, or $2\le |\pi (G)|\le 3$; if $|\pi (G)|=3$, then $G$ has a Sylow tower of supersoluble type; if $|\pi (G)|=2$, then either $G$ has a normal Sylow subgroup or, for the largest $p\in \pi (G)$, some maximal subgroup of a Sylow $p$-subgroup is normal in $G$. If $G$ is nonsoluble and, for each maximal subgroup of $G$, there exists a supersoluble subgroup of the same order, then every nonabelian composition factor of $G$ is isomorphic to $PSL_2(p)$ for some prime $p$; we list all such values of $p$.