  We show that given a compact group $G$ acting continuously on a metric space
$M$ by bi-Lipschitz bijections with uniformly bounded norms, the Lipschitz-free
space over the space of orbits $M/G$ (endowed with Hausdorff distance) is
complemented in the Lipschitz-free space over $M$. We also investigate the more
general case when $G$ is amenable, locally compact or SIN and its action has
bounded orbits. Then we get that the space of Lipschitz functions $Lip_0(M/G)$
is complemented in $Lip_0(M)$. Moreover, if the Lipschitz-free space over $M$,
$F(M)$, is complemented in its bidual, several sufficient conditions on when
$F(M/G)$ is complemented in $F(M)$ are given. Some applications are discussed.
The paper contains preliminaries on projections induced by actions of amenable
groups on general Banach spaces.
