  We prove a Fatou-type theorem and its converse for certain positive
eigenfunctions of the Laplace-Beltrami operator $\mathcal{L}$ on a Harmonic
$NA$ group. We show that a positive eigenfunction $u$ of $\mathcal{L}$ with
eigenvalue $\beta^2-\rho^2$, $\beta\in (0,\infty)$, has an admissible limit in
the sense of Kor\'anyi, precisely at those boundary points where the strong
derivative of the boundary measure of $u$ exists. Moreover, the admissible
limit and the strong derivative are the same. This extends a result of Ramey
and Ullrich regarding nontangential convergence of positive harmonic functions
on the Euclidean upper half space.
