  For symplectic group actions which are not Hamiltonian there are two ways to
define reduction. Firstly using the cylinder-valued momentum map and secondly
lifting the action to any Hamiltonian cover (such as the universal cover), and
then performing symplectic reduction in the usual way. We show that provided
the action is free and proper, and the Hamiltonian holonomy associated to the
action is closed, the natural projection from the latter to the former is a
symplectic cover. At the same time we give a classification of all Hamiltonian
covers of a given symplectic group action. The main properties of the lifting
of a group action to a cover are studied.
