Note on the induced metric on power sets of groups

In this note we clarify general properties of the Hausdorff-like metric on the power set $\mathcal{S}(G)$ of a group $G$ induced from word length norm and obtain some results on quasi-isometries between some subspaces of $\mathcal{S}(G)$ and $\mathcal{S}(H)$ for a group epimorphism $f:G\to H$, when $\mathcal{S}(\mathrm{K}\mathrm{e}\mathrm{r}f)$ is metrically bounded.