On strictly $2$-maximal subgroups of finite groups

We give examples of finite soluble and simple groups in which every $2$-maximal subgroup is strictly $2$-maximal. We prove that if in a group $G$ there is a strictly $2$-maximal subgroup of order $2$, then $G$ is a supersoluble group of order $2pq$, where $p$ and $q$ are primes, not necessarily distinct, or $G$ is isomorphic to the alternating group $A_4$. We establish the structure of a finite group in which every $2$-maximal subgroup is a Hall subgroup. We prove that the requirement of $\mathfrak{F}$-subnormality of all strictly $2$-maximal subgroups coincides with the requirement of subnormality of all $2$-maximal subgroups of a group $G$ for a subgroup-closed saturated lattice formation $\mathfrak{F}$ containing all nilpotent groups and $G\notin\mathfrak{F}$.