Denote by $F_n$ the free group of rank $n$. We say that an automorphism $\phi\in Aut(F_n)$ is *geometric* if there exists a surface with boundary $M$ and a homeomorphism $h\colon M\to M$ such that $h$ induces $\phi$ on $\pi_1$.

Every automorphism of $F_2$ is geometric, but in higher rank geometric automorphisms are rare. I have seen several examples of nongeometric automorphisms, but all of which were of infinite order.

**Question:** Are there any examples of *periodic* (i.e. finite-order) nongeometric automorphism? Or can one show that periodic automorphisms are always geometric, possibly making use of the fact that every periodic automorphism can be realized as a simplicial graph automorphism? I wouldn't be surprised if we could nicely embed the graph into a suitable surface and extend the automorphism of the graph to a surface homeomorphism.

**Observation:** Every periodic automorphism $\phi$ fixes some nontrivial conjugacy class (maybe up to inversion), which is a necessary condition for $\phi$ to possibly be induced by a homeomorphism of a surface with *connected* boundary. But this observation really only plays a role when $n$ is even, as the fundamental group of a surface with connected boundary has even rank.