Finite simple groups that are not spectrum critical

Let $G$ be a finite group. The spectrum of $G$ is the set $\omega(G)$ of orders of all its elements. The subset of prime elements of $\omega(G)$ is called prime spectrum and is denoted by $\pi(G)$. A group $G$ is called spectrum critical ( prime spectrum critical) if, for any subgroups $K$ and $L$ of $G$ such that $K$ is a normal subgroup of $L$, the equality $\omega(L/K)=\omega(G)$ ($\pi(L/K)=\pi(G)$, respectively) implies that $L=G$ and $K=1$. In the present paper, we describe all finite simple groups that are not spectrum critical. In addition, we show that a prime spectrum minimal group $G$ is prime spectrum critical if and only if its Fitting subgroup $F(G)$ is a Hall subgroup of $G$.