In P. Deligne. Extensions centrales non résiduellement finies de groupes
arithmétiques. CR Acad. Sci. Paris, série A-B, 287, 203–208, 1978. Deligne proves the existence of a certain central extension of a residually finite group.
See the section Deligne's central extension in Cornulier-Guyot-Pitsch's paper for a quick discussion of the group.
A countable, discrete group $\Gamma$ is sofic if for every $\epsilon>0$ and finite subset $F$ of $\Gamma$ there exists an $(\epsilon,F)$-almost action of $\Gamma$. See, for example Theorem 3.5 of the nice survey of Pestov. Gromov asked whether all countable discrete groups are sofic. It is now widely believed that there should be a counterexample to this.
Is Deligne's central extension sofic?
This question is related to the one here, but is not sharpened enough to be an answer. (In fact, in its original form not even to be a question! Thank you Henry and Andreas.)
Best Answer
If I understand your question correctly then I think you've already answered it! The 'above property' is precisely 'not being residually finite'. To see this, just consider the intersection of all finite-index subgroups: if it's trivial, your group is residually finite; if not, then that intersection is your subgroup $H$.
As we found in a previous question of yours, Baumslag's group
$B=\langle a,b\mid (a^b)^{-1}a(a^b)a^{-2}\rangle$
which is certainly not residually finite, is sofic.