Bi-elliptic Weierstrass points on curves of genus 5

We study genus 5 curves with three bi-elliptic involutions by relating them to certain genus 3 curves. In particular, we show that there is only one such curve having exactly 24 Weierstrass points. This corrects a paper of del Centina. We also classify all genus 5 curves whose automorphism group is transitive on the Weierstrass points.