Let $D=(V,A)$ be a finite directed graph, and suppose that

• $D$ is vertex-transitive,
• $D$ is edge-transitive, and
• between any two vertices there is at most one edge, in particular, if $(v,w)\in A$ then $(w,v)\notin A$.

Question: Is it true that every non-identity involution $\phi\in\operatorname{Aut}(D)$ (i.e. $\phi^2=\operatorname{id}$) can be written as $\phi=\sigma^n$ for some $\sigma\in\operatorname{Aut}(D)\setminus\{\phi\}$ and $n\ge 2$?

For example, consider the directed cycle below. The 180° rotation is the only involution, and it can be written as three times the application of a 60° rotation.

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