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 ${\cal S}(G)$ of a group $G$ induced from word length norm and obtain some results on quasi-isometries between some subspaces of ${\cal S}(G)$ and ${\cal S}(H)$ for a group epimorphism $f : G \to H$, when ${\cal S}({\rm Ker}\,f)$ is metrically bounded.