  Symbolic substitution systems are an important source of aperiodic Delone
sets in abelian locally compact groups. In this article, we consider a large
class of non-abelian nilpotent Lie groups with dilation structures, which we
refer to as rationally homogeneous Lie groups with rational spectrum
(RAHOGRASPs). We show, by explicit construction, that every RAHOGRASP admits a
lattice with a primitive symbolic substitution system and hence contains a
weakly aperiodic linearly repetitive Delone set. These are the first examples
of weakly aperiodic linearly repetitive Delone sets in non-abelian Lie groups.
Building on our previous work, we establish unique ergodicity of the
corresponding Delone dynamical systems. Our construction applies in particular
to all two-step nilpotent Lie groups defined over $\mathbb Q$ such as the
Heisenberg group. In this case, the Delone sets in question are in fact
strongly aperiodic, i.e. the underlying action of the corresponding Delone
dynamical system is free.
