  We consider left-invariant distances $d$ on a Lie group $G$ with the property
that there exists a multiplicative one-parameter group of Lie automorphisms
$(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so
that $ d(\delta_\lambda x,\delta_\lambda y) = \lambda d(x,y)$, for all $x,y\in
G$ and all $\lambda>0$.
  First, we show that all such distances are admissible, that is, they induce
the manifold topology. Second, we characterize multiplicative one-parameter
groups of Lie automorphisms that are dilations for some left-invariant distance
in terms of algebraic properties of their infinitesimal generator.
  Third, we show that an admissible left-invariant distance on a Lie group with
at least one nontrivial dilating automorphism is biLipschitz equivalent to one
that admits a one-parameter group of dilating automorphisms. Moreover, the
infinitesimal generator can be chosen to have spectrum in $[1,\infty)$. Fourth,
we characterize the automorphisms of a Lie group that are a dilating
automorphisms for some admissible distance.
  Finally, we characterize metric Lie groups admitting a one-parameter group of
dilating automorphisms as the only locally compact, isometrically homogeneous
metric spaces with metric dilations of all factors. Such metric spaces appear
as tangents of doubling metric spaces with unique tangents.
