  For a connected Lie group $G$ and an automorphism $T$ of $G$, we consider the
action of $T$ on Sub$_G$, the compact space of closed subgroups of $G$ endowed
with the Chabauty topology. We study the action of $T$ on Sub$^p_G$, the
closure in Sub$_G$ of the set of closed one-parameter subgroups of $G$. We
relate the distality of the $T$-action on Sub$^p_G$ with that of the $T$-action
on $G$ and characterise the same in terms of compactness of the closed subgroup
generated by $T$ in Aut$(G)$ when $T$ acts distally on the maximal central
torus and $G$ is not a vector group. We extend these results to the action of
subgroups ${\mathcal H}$ of Aut$(G)$ and equate the distal action of any closed
subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal
H}$. Moreover, we show that a connected Lie group $G$ acts distally on
Sub$^p_G$ by conjugation if and only if $G$ is either compact or it is
isomorphic to a direct product of a compact group and a vector group. Our
results generalise some results proven by Shah and Yadav.
