I am analysing involutions classes in $S_8(2)=O_9(2)$ and $O_{10}^-(2)$. The smallest conjugacy class $2A$ in the first group is of size $255$. In the second group we can find "small" conjugacy class of size $528$ in the extension $O_{10}^-(2).2$. I wonder whether there is any explanation for this phenomen that sometimes smallest involution conjugacy class can be found in simple group $G$ and sometimes in the extension $G.2$. Another obvious example is $A_n$ and extension $S_n=A_n.2$ containing transposition class.
Another guess I made is that in each simple finite group there exists involution conjugacy class with odd order. Is this something known ? Again, I wonder when smallest involution conjugacy class is of odd order.