On the quasi-$\mathfrak{F}$-hypercenter of a finite group

In this paper some properties of the ${\mathfrak{F}}^{\ast }$-hypercenter of a finite group are studied where ${\mathfrak{F}}^{\ast }$ is the class of all finite quasi-$\mathfrak{F}$-groups for a hereditary saturated formation $\mathfrak{F}$ of finite groups. In particular, it is shown that the quasinilpotent hypercenter of a finite group coincides with the intersection of all maximal quasinilpotent subgroups.