Some questions on quasinilpotent groups and related classes

Abstract

In this paper we will prove that if $G$ is a finite group, $X$ a subnormal subgroup of $X F^*(G)$ such that $X F^*(G)$ is quasinilpotent and $Y$ is a quasinilpotent subgroup of $N_G(X)$, then Y F^*(N_G(X}) is quasinilpotent if and only if $Y F^*(G)$ is quasinilpotent. Also we will obtain that $F^*{G}$ controls its own fusion in $G$ if and only if $G=F^*{G}$.