  We show that an approximate lattice in a nilpotent Lie group admits a
relatively dense subset of central $(1-\epsilon)$-Bragg peaks for every
$\epsilon > 0$. For the Heisenberg group we deduce that the union of horizontal
and vertical $(1-\epsilon)$-Bragg peaks is relatively dense in the unitary
dual. More generally we study uniform approximate lattices in extensions of
lcsc groups. We obtain necesary and sufficient conditions for the existence of
a continuous horizontal factor of the associated hull-dynamical system, and
study the spectral theory of the hull-dynamical system relative to this
horizontal factor.
