Coarse embeddings of quotients by finite group actions

We prove that for a metric space $X$ and a finite group $G$ acting on $X$ by isometries, if $X$ coarsely embeds into a Hilbert space, then so does the quotient $X\mathrm{/}G$. A crucial step towards our main result is to show that for any integer $k>0$ the space of unordered $k$-tuples of points in Hilbert space, with the $1$-Wasserstein distance, itself coarsely embeds into Hilbert space. Our proof relies on establishing bounds on the sliced Wasserstein distance between empirical measures in ${\mathbb{R}}^n$.