  Let $N$ be a simply connected, connected non-commutative nilpotent Lie group
with Lie algebra $\mathfrak{n}$ of dimension $n.$ Let $H$ be a subgroup of the
automorphism group of $N.$ Assume that $H$ is a commutative, simply connected,
connected Lie group with Lie algebra $\mathfrak{h}.$ Furthermore, let us assume
that the linear adjoint action of $\mathfrak{h}$ on $\mathfrak{n}$ is
diagonalizable with non-purely imaginary eigenvalues. Let
$\tau=\mathrm{Ind}%_{H}^{N\rtimes H} 1$. We obtain an explicit direct integral
decomposition for $\tau$, including a description of the spectrum as a
sub-manifold of $(\mathfrak{n}+\mathfrak{h})^{\ast}$, a formula for the
multiplicity function of the unitary irreducible representations occurring in
the direct integral, and a precise intertwining operator. Finally, we
completely settle the admissibility question of $\tau$. In fact, we show that
if $G=N\rtimes H$ is unimodular, then $\tau$ is never admissible, and if $G$ is
nonunimodular, $\tau$ is admissible if and only if the intersection of $H$ and
the center of $G$ is equal to the identity of the group.
