Let $H \subset \mathrm{Sp}(\mathbb{Z})$ be a subgroup of the simplectic group of (square of even dimension) matrices with integer entries, and let $H^{(\ell)}$ denote its pro-$\ell$ completion for some prime $\ell$. Then does $H$ being Zariski dense in $\mathrm{Sp}(\mathbb{Z})$ imply that the abelianization $(H^{(\ell)})^{ab}$ is finite?

Or does the converse hold (i.e. $(H^{\ell})^{ab}$ is finite for some $\ell$ implies $H < \mathrm{Sp}(\mathbb{Z})$ is Zariski dense)?

I have no idea how to go about solving this, and half expect that there are easy counterexamples that I'm not thinking of.