Conjugacy class of involutions in finite simple group

Question

I've been working on this problem, but I can't seem to do it.

For a finite simple group $G$, is there a unique conjugacy class consisting of involutions?

Clearly, a conjugacy class with an involution also contains other involutions, but could there possibly be another conjugacy class containing involutions?

What if we restrict $G$ to be $A_n$ (the alternating group on $n$ symbols)? Is there a unique conjugacy class consisting of involutions?

Answer 1

$A_8$ has two conjugacy classes of involutions, one with cycle structure $2^2\,1^4$ and the other with cycle structure $2^4$.