On strongly $\Pi$-permutable subgroups of a finite group

Let $\sigma=\{\sigma_i\mid i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, let $\varnothing\ne\Pi\subseteq\sigma$, and let $G$ be a finite group. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\Pi$-set of $G$ if each member $\ne 1$ of $\mathcal{H}$ is a Hall $\sigma_i$-subgroup of $G$ for some $\sigma_i\in\Pi$  and $\mathcal{H}$ has exactly one Hall $\sigma_i$-subgroup of $G$ for every $\sigma_i\in\Pi$  such that $\sigma_i\cap\pi(G)\ne\emptyset$. A subgroup $A$ of $G$ is called (i) $\Pi$-permutable in $G$ if $G$ has a complete Hall $\Pi$-set $\mathcal{H}$ such that $AH^x=H^xA$ for all $H\in\mathcal{H}$ and $x\in G$; (ii) $\sigma$-subnormal in $G$ if there is a subgroup chain $A=A_0\leqslant A_1\leqslant\dots\leqslant A_t=G$ such that either $A_{i-1}\leqslant A_i$ or $A_i/(A_{i-1})A_i$ is a $\sigma_k$-group for some $k$ for all $i=1,\dots,t$; and (iii) strongly $\Pi$-permutable if $A$ is $\Pi$-permutable and $\sigma$-subnormal in $G$. We study the strongly $\Pi$-permutable subgroups of $G$. In particular, we give characterizations of these subgroups and prove that the set of all strongly $\Pi$-permutable subgroups of $G$ forms a sublattice of the lattice of all subgroups of $G$.