  Since Gruenbaum and Shephard's investigation of self-intersection-free
polyhedra with positive genus and vertex-transitive symmetry in 1984 the
question of complete classification of such objects in Euclidean 3-space has
been open. Due to a recent article by Gevay, Schulte, and Wills, we now know
that the genus range g>=2 can only admit a finite number of vertex-transitive
polyhedra, all with rotational Platonic symmetry. In this article, we show that
the symmetry group must also act simply transitively on the vertices.
Furthermore, the case of rotational tetrahedral symmetry is settled completely
- the unique example is already known.
