Hi everyone!

Given a finite simple group $S$, is it always possible to find two elements $x,y \in S$ with the property that for every $a,b \in S$ we have $\langle x^a,y^b \rangle = S$?

More in general, given a finite almost simple group $X$ with socle $S$, is it possible to find two elements $x,y \in X$ with the property that for every $a,b \in X$ we have $\langle x^a,y^b \rangle \supseteq S$?

I can deal with the alternating case, but I don't know where to look for results about the general case.

Thank you for any contribution.