= Solution
Let $S$ be a finite generating set of $H$, choose one lift $\widetilde s\in G$ for every $s\in S$, and put $K=\ker f$. The finite set
$$
\widetilde S\cup K
$$
generates $G$: lift a word representing $f(g)$ and observe that the discrepancy from $g$ lies in $K$.
Use this generating set for $G$. The quotient map does not increase word length, while lifting a shortest word in $H$ leaves only a final element of $K$, whose word length is at most one. Hence
$$
d_H(f(g),f(g'))
\le d_G(g,g')
\le d_H(f(g),f(g'))+1.
$$
The map is surjective, so it is a <finite-kernel quotient quasi-isometry>. Therefore $G$ is finitely generated and quasi-isometric to $H$.
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