One-photon single ionization of molecules has been at the focus of several discussions concerning the reconstruction of observables obtained with ultrashort pulses from those obtained from continuous wave radiation (and vice versa). A related controversy on the conditions and observables that allow for coherent control in one-photon processes has been recently revisited (Science 2006, 313, 1257; J. Chem. Phys. 2010, 133, 151101). Our benchmark to investigate these issues is photoionization of the hydrogen molecule, where the autoionization events are the time-dependent processes in field-free evolution that could serve as a target for coherent control. We show that the variation of one-photon ionization probabilities with pulse duration are solely due to spectral effects and thus cannot be coherently controlled. We then discuss for which observables and under which conditions phase control of autoionization dynamics is possible.