= Solution
The <Milnor–Švarc lemma> applies to an isometric <properly discontinuous group action> that is also a <cocompact group action> on a nonempty <proper metric space> that is a <geodesic metric space>. It concludes that the group is finitely generated and, for a basepoint $x$ and any finite <generating set of a group> $S$, the <orbit map>
$$
\boxed{(G,d_S)\longrightarrow X,\qquad g\longmapsto gx,\text{ is a quasi-isometry}.}
$$
Here $d_S$ is the <word metric>, and a <quasi-isometry> means a map satisfying two-sided coarse distance bounds with a uniformly coarsely dense image. A free action is not required: finite stabilizers are compatible with proper discontinuity. The properness and cocompactness hypotheses ensure that only finitely many group elements move a chosen basepoint by the bounded amount used to generate the group. Replacing a finite generating set by another does not change the conclusion, by <equivalence of finite word metrics>.
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