Consider a central extension of groups $\mathcal{E}$ :
$$1\rightarrow A\rightarrow \widetilde{G}\rightarrow G \rightarrow 1$$with $A$ finite cyclic. Does there always exist a finite index subgroup $H$ of $G$ such that the restriction of $\mathcal{E}$ to $H$ becomes trivial? 



Does a central extension split over some finite index subgroup?
No. Deligne gave a famous example of a central extension
$$
1\to\mathbb{Z}/2\to \widetilde{G}\to G\to 1
$$
such that $G$ is a finite-index subgroup of $PSp(2n,\mathbb{Z})$ but $\widetilde{G}$ is not residually finite. (See, for instance, the references in [this](https://mathoverflow.net/questions/48656/is-delignes-central-extension-sofic) MO question.) But it's easy to see that if the extension were to virtually split then $\widetilde{G}$ would be residually finite.


