It appears to be standard that the set of non-identity involutions in $SL(2, 2^n) = PSL(2, 2^n)$ forms a single conjugacy class. What is the best reference for this?

I note that https://math.stackexchange.com/questions/208255/conjugacy-classes-of-elements-of-a-prime-order-in-psl-2q asked a related question and seems to contain a proof, but is there a published proof?