Element orders and the structure of a finite group

To every finite group $G$, we can assign the set $\omega(G)$ consisting of all positive integers arising as element orders of $G$ (so, for example, $\omega(A_5)=\{1,2,3,5\}$). It is a natural question to ask what we can say about the structure of $G$ given some properties of $\omega(G)$. Within this framework, I will discuss a more narrow question of to what extent $\omega(G)$ determines $G$ provided that $G$ is a finite nonabelian simple group.