Hi, I am troubled with the following question: Does there exist a finite order automorphism of a free group, $f\in Aut(F)$, such that it fixes no non trivial conjugacy class and no non trivial centralizer, i.e. $f(g)$ is not conjugate to $g$ and $f(g)\neq g^{-1}$ for any $g\in F$ ? Can we find such, so the same holds for any of its non trivial powers $f^i$?

By inspection of the finite order automorphisms of $F_2$, this cannot be an automorphism of $F_2$.

Moreover, if there exists such an automorphism is it possible to find one with as large order as we want (possibly after moving to a larger free group)?