The recurrence of a lazy random walk is related to its subrandom walk.

Question

Let $(X_n)$ be a lazy random walk on a finitely generated group. Show that for any $m \in \mathbb{N}$, the random walk $(X_{mn})$ is recurrent if and only if $(X_n)$ is recurrent. It is easy to show $(X_{n})$ is recurrent if $(X_{mn})$ is recurrent, but how to prove the converse?