  For a locally compact metrizable group $G$, we study the action of $\rm
Aut(G)$ on $\rm Sub_G$, the set of closed subgroups of $G$ endowed with the
Chabauty topology. Given an automorphism $T$ of $G$, we relate the distality of
the $T$-action on $\rm Sub_G$ with that of the $T$-action on $G$ under a
certain condition. If $G$ is a connected Lie group, we characterise the
distality of the $T$-action on $\rm Sub_G$ in terms of compactness of the
closed group generated by $T$ in $\rm Aut(G)$ under certain conditions on the
center of $G$ or on $T$ as follows: $G$ has no compact central subgroup of
positive dimension or $T$ is unipotent or $T$ is contained in the connected
component of the identity in $\rm Aut(G)$. Moreover, we also show that a
connected Lie group $G$ acts distally on $\rm Sub_G$ if and only if $G$ is
either compact or it is isomorphic to a direct product of a compact group and a
vector group. All the results on the Lie groups mentioned above hold for the
action on $\rm Sub^a_G$, a subset of $\rm Sub_G$ consisting of closed abelian
subgroups of $G$.
