On one operation on the formations of finite groups

Let $\pi$ be a set of primes. In this article, the operation $w_\pi^*$ on the formations of finite groups is introduced. If $\mathfrak{F}$ is a non-empty formation, then $w_\pi^*\mathfrak{F}$ is the class of all groups $G$ such that $\pi(G)\subseteq\pi(\mathfrak{F})$ and every Sylow $q$-subgroup of $G$ is strongly $\mathrm{K}$-$\mathfrak{F}$-subnormal in $G$ for $q\in\pi\cap\pi(G)$. The properties of $w_\pi^*$ are obtained, in particular, $w_\pi^*\mathfrak{F}=w_\pi^*(w_\pi^*\mathfrak{F})$ for hereditary formations $\mathfrak{F}$. Hereditary saturated formations $\mathfrak{F}$ for which $w_\pi^*\mathfrak{F}$ coincides with $\mathfrak{F}$ have been found.