We study the wake of a cylinder performing rotary oscillations around its axis at moderate Reynolds number. We observe that the structure of the vortex shedding is strongly affected by the forcing parameters. The forced wake is characterized by a ‘lock-in’ region where the vortices are shed at the forcing frequency and a region where the vortices can be reorganized to give a second frequency close to those observed for the unforced wake. We show that these modifications of the wake structure change the dynamic of the fluctuations downstream from the cylinder. We vary the amplitude and the frequency of the oscillations and study the consequences of these modifications on the mean flow and the global drag applied on the cylinder. We then discuss the mechanism responsible for the modification of the fluctuations and the modification of the drag coefficient.