  We prove $L^p\to L^q$ estimates for the local maximal operator associated
with dilates of the K\'oranyi sphere in Heisenberg groups. These estimates are
sharp up to endpoints and imply new bounds on sparse domination for the
corresponding global maximal operator. We also prove sharp $L^p\to L^q$
estimates for spherical means over the Kor\'anyi sphere, which can be used to
improve the sparse domination bounds for the associated lacunary maximal
operator.
