I only consider convex polytopes, i.e. convex hulls of finitely many points. The (edge-)graph of a polytope $ P\subseteq\Bbb R^d$ is the graph consisting of the polytope’s vertices, two are adjacent if they lie on a common edge in $ P$ .
The affine symmetry group of $ P$ induces a permutation symmetry group on its graph. We say that a polytope is vertex-, edge-, arc– and/or half-transitive if the symmetry group induced on its graph acts in the respective way.
Question: Are there half-transitive polytopes, i.e. polytopes which are vertex- and edge-transitive, but not arc-transitive?
The graph of such a polytope must be half-transitive, and such graphs are rare (the smallest one is the 4-valent Holt graph on 27 vertices).
I looked a bit into chiral polytopes (i.e. two flag orbits), but I read somewhere that these only exist for abstract polytopes, and not for convex ones. However, I think that chirality is a much stronger requirement for a polytope than half-transitivity (as the latter only speaks about vertices and edges instead of flags).