Lacunary maximal functions on homogeneous groups

We observe that classical arguments of Ricci–Stein can be used to prove $L^p$ bounds for maximal functions associated to lacunary dilates of a fixed measure in the setting of homogenous groups. This recovers some recent results on averages over Korányi spheres and horizontal spherical averages of a type introduced by Nevo–Thangavelu. Moreover, the main theorem applies much more broadly and we explore its consequences through a variety of explicit examples.