Simple groups with normal generating sets

Question

Is it known that which finite non-abelian simple groups have a minimal generating set $S$ which is normal in $G$ (i.e. for every $g\in G$, we have $gSg^{-1}=S$)?

Answer 1

I understand that "minimal" means "minimal for inclusion among subsets generating $G$ as a group".

Fact: No nonabelian group (finite or not, simple or not) has a minimal generating subset that is conjugacy-closed.

Indeed, let a nonabelian group $G$ have such a subset $S$. Since $G$ is nonabelian, there are two non-commuting elements $s,t$ in $S$. Then $sts^{-1}\in S$, since $S$ is conjugacy-closed. Also $sts^{-1}\in\{s,t\}$ is excluded because each implies that $s$ and $t$ commute. Hence, $sts^{-1}$ is a redundant generator, contradiction.