Infinitely many Riemann surfaces with a transitive action on the Weierstrass points

In this short note, we prove the existence of infinitely many pairwise nonisomorphic, non-hyperelliptic Riemann surfaces with automorphism group acting transitively on the Weierstrass points. We also find all compact Riemann surfaces with automorphism group acting transitively on the Weierstrass points, under the assumption that they are simple.