  In this paper we deal with lacunary and full versions of the spherical
maximal function on the Heisenberg group $\mathbb{H}^n$, for $n\ge 2$. By
suitable adaptation of an approach developed by M. Lacey in the Euclidean case,
we obtain sparse bounds for these maximal functions, which lead to new
unweighted and weighted estimates. In particular, we deduce the $L^p$
boundedness, for $1<p<\infty$, of the lacunary maximal function associated to
the spherical means on the Heisenberg group. In order to prove the sparse
bounds, we establish $ L^p-L^q $ estimates for local (single scale) variants of
the spherical means.
