  In this paper we investigate the $L^p$ boundedness of the lacunary maximal
function $ M_{\Ha}^{lac} $ associated to the spherical means $ A_r f$ taken
over Koranyi spheres on the Heisenberg group. Closely following an approach
used by M. Lacey in the Euclidean case, we obtain sparse bounds for these
maximal functions leading to new unweighted and weighted estimates. The key
ingredients in the proof are the $L^p$ improving property of the operator
$A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where
$\tau_yf(x)=f(xy^{-1})$ is the right translation operator.
