  For a locally compact group $G$, we study the distality of the action of
automorphisms $T$ of $G$ on ${\rm Sub}_G$, the compact space of closed
subgroups of $G$ endowed with the Chabauty topology. For a certain class of
discrete groups $G$, we show that $T$ acts distally on ${\rm Sub}_G$ if and
only if $T^n$ is the identity map for some $n\in{\mathbb N}$. As an
application, we get that for a $T$-invariant lattice $\Gamma$ in a simply
connected nilpotent Lie group $G$, $T$ acts distally on ${\rm Sub}_G$ if and
only if it acts distally on ${\rm Sub}_\Gamma$. This also holds for any closed
$T$-invariant co-compact subgroup $\Gamma$. For a lattice $\Gamma$ in a simply
connected solvable Lie group, we study conditions under which its automorphisms
act distally on ${\rm Sub}_\Gamma$. We construct an example highlighting the
difference between the behaviour of automorphisms on a lattice in a solvable
Lie group from that in a nilpotent Lie group. For torsion-free compactly
generated nilpotent (metrizable) groups $G$, we obtain the following
characterisation: $T$ acts distally on ${\rm Sub}_G$ if and only if $T$ is
contained in a compact subgroup of ${\rm Aut}(G)$. Using these results, we
characterise the class of such groups $G$ which act distally on ${\rm Sub}_G$.
We also show that any compactly generated distal group $G$ is Lie projective.
As a consequence, we get some results on the structure of compactly generated
nilpotent groups.
