On the influence of the $\mathfrak{F}$-hypercenter on the structure of finite multifactorized groups

The structure of a finite group having $t$ (normal, abnormal) subgroups with pairwise mutually coprime indices and $\mathfrak{F}$-hypercentrality constraints on pairwise intersections of these subgroups, where $\mathfrak{F}$ is a saturated formation, is described. As a consequence, new characterizations of $\Sigma_t$-closed, weakly $R_t$-closed, and $N_0$-closed formations of finite groups are established. The obtained results make a significant contribution to the solution of the question posed by W. Guo and A.N. Skiba in 2011.