Extending finite free actions of surfaces

We prove the existence of finite groups acting freely as orientation‐preserving homeomorphisms on closed orientable surfaces that extend as a group of homeomorphisms of some compact orientable 3‐manifold but that cannot extend to a handlebody. This solves a basic problem in low‐dimensional equivariant topology going back to the work of Reni and Zimmermann in the mid‐1990s.