An approach for the control of a class of underactuated mechanical systems on Lie groups, including many systems previously studied in the control literature, like underactuated planar manipulators and rigid bodies (spacecrafts, hovercrafts, etc), is proposed. The main outcome of the paper is the derivation, based on the transverse function approach initially proposed by the authors for the control of non-holonomic (driftless) mechanical systems, of smooth feedback control laws which stabilize, in a practical sense, any (possibly non-admissible) reference trajectory in the configuration space.