  We study by Quantum Monte Carlo simulations the phase diagram of lattice hard
core bosons with nearest-neighbour repulsive interactions, in the presence of a
super-lattice of adsorption sites. For a moderate adsorption strength, the
system forms crystal phases registered with the adsorption lattice; a
"supersolid" phase exists, on both the vacancy and interstitial sides, whereas
at commensuration the superfluid density vanishes. The possible relevance of
these results to experiments on $^4$He films adsorbed on graphite is discussed.
