On an open problem about $\pi$-quasi-$\mathfrak{F}$-groups

Abstract

Let $\mathfrak{F}$ be a class of finite groups, $p$ a prime and $\pi$ a set of some primes. A finite group $G$ is called a $p$-quasi-$\mathfrak{F}$-group (respectively, by $\pi$-quasi-$\mathfrak{F}$-group) provided that for every $\mathfrak{F}$-eccentric $G$-chief factor $H/K$ of order divisible by $p$ (respectively, by at least one prime in $\pi$), the automorphisms of $H/K$ induced by all elements of $G$ are inner. In this paper, we obtain the characterizations of $p$-quasi-$\mathfrak{F}$-groups and $\pi$-quasi-$\mathfrak{F}$-groups, which give a positive answer to an open problem in the book [3].