  We study the geometry of Lie groups $G$ with a continuous Finsler metric,
assuming the existence of a subgroup $K$ such that the metric is
right-invariant for the action of $K$. We present a systematic study of the
metric and geodesic structure of homogeneous spaces $M$ obtained by the
quotient $M\simeq G/K$. Of particular interest are left-invariant metrics of
$G$ which are then bi-invariant for the action of $K$. We then focus on the
geodesic structure of groups $K$ that admit bi-invariant metrics, proving that
one-parameter groups are short paths for those metrics, and characterizing all
other short paths. We provide applications of the results obtained, in two
settings: manifolds of Banach space linear operators, and groups of maps from
compact manifolds.
