On the intersections of generalized projectors with the products of normal subgroups of finite groups

The factorization properties of the $\mathfrak{F}^\omega$-projector introduced by V. A. Vedernikov and M. M. Sorokina in 2016 ($\omega$ is a non-empty set of primes and $\mathfrak{F}$ is a non-empty class of groups) were investigated. Necessary and sufficient conditions are found for the equality $N_1N_2 \cap H = (N_1 \cap H)(N_2 \cap H)$ for any $\mathfrak{F}^\omega$-projector $H$ and any normal $\omega$-subgroups $N_1$ and $N_2$ of $G$, where $G$ is an extension of the $\omega$-group with the help of an $\mathfrak{F}$-group.