  Our aim in this article is to study the weighted boundedness of the centered
Hardy-Littlewood maximal operator in Harmonic $NA$ groups. Following Ombrosi et
al. \cite{ORR}, we define a suitable notion of $A_p$ weights, and for such
weights, we prove the weighted $L^p$-boundedness of the maximal operator.
Furthermore, as an endpoint case, we prove a variant of the Fefferman-Stein
inequality, from which vector-valued maximal inequality has been established.
We also provide various examples of weights to substantiate many aspects of our
results. In particular, we have shown certain spherical functions of the
Harmonic $NA$ group constitute examples of $A_p$ weights. The purely
exponential volume growth property of the Harmonic $NA$ group has played a
crucial role in our proofs.
