We investigate the stability of the steady vertical path and the emerging trajectories of a buoyancy -driven annular disk as the diameter of its central hole is varied. The steady and axisymmetric wake associated with the steady vertical path of the disk, for small hole diameters, behaves similarly to the one past a permeable disk, with the detachment of the vortex ring due to the bleeding flow through the hole. However, as the hole diameter increases, a second recirculating vortex ring of opposite vorticity forms at the internal edge of the annulus. A further increase in the hole size leads to the shrinking of these recirculating regions until they disappear. The flow modifications induced by the hole influence the stability features of the steady and axisymmetric flow associated with the steady vertical path. The fluid -solid coupled problem shows a nonmonotonic behavior of the critical Reynolds number for the destabilization of the steady vertical path, for low values of the disk's moment of inertia. However, for very large holes, with dimension approximately more than half of the external diameter, a marked increase of the neutral stability threshold is observed. The nature of the primary instability changes as the hole size increases, with large (small) amplitude oscillations of the trajectory at intermediate (very small and large) internal diameters. We then illustrate results obtained with fully nonlinear simulations of the time -dependent dynamics, together with a comparison of the linear stability analysis results. Falling styles, typically described as steady, hula -hoop, fluttering, chaotic, and tumbling, are shown to emerge as attractors for the nonlinear dynamics of the coupled fluid -structure system. The presence of a central hole does not always reduce the falling Reynolds number, and it may cause the transition from tumbling towards fluttering, from fluttering to hula -hoop, and from hula -hoop to steady, hence promoting trajectories with smaller lateral deviations from the vertical path. The observed trajectories and patterns agree well with linear stability analysis results, in the vicinity of the threshold of instability.