For $m \geq 2$, let $(\mathbb{ Z}_{ m+1}^N, |\cdot|)$ denote the group equipped with the so-called $l^0$ metric, \[ |y| = \left| \big( y(1), \dots, y(N) \big) \right| := | \{ 1 \leq i \leq N : y(i) \neq 0 \} |,\] and define the $L^1$-normalized indicator of the $r$-sphere, \[ \sigma_r := \frac{ 1}{ |\{ |x| = r\}|} 1_{ \{ |x| =r\}}.\] We study the $L^p \to L^p$ mapping properties of the maximal operator \[ M^{ N} f (x) := \sup_{ r \leq N} | \sigma_r*f| \] acting on functions defined on $\mathbb{ Z}_{ m+1}^N$. Specifically, we prove that for all $p>1$, there exist absolute constants $C_{ m,p}$ so that \[ \| M^{ N} f \|_{ L^p(\mathbb{ Z}_{ m+1}^N)} \leq C_{ m,p} \| f \|_{ L^p(\mathbb{ Z}_{ m+1}^N)} \] for all $N$.
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