I can construct a finitely presented group $G$ with the following property (which I use to construct something else).

Given a finitely preseted group $\Gamma$, there is a subgroup $G'\le G$ of finite index such that $$\Gamma=G'/\langle\mathrm{Tor}\, G'\rangle ,$$ where $\mathrm{Tor}\, G'\subset G'$ is the set of all elements of finite order.

I think to call such group $G$ *universal*.

**Questions:**

- Was it already constructed?
- Does it already has a name? Is there any closely related terminology?

**P.S.**

- The group which I construct is in fact hyperbolic.
- The construction is simple, but it takes 2--3 pages. Let me know if you see a short way to do it.
- Here, the term "universal group" was used in very similar context (thanks to D. Panov for the reference).
- Thanks to all your comments, we call them "telescopic" actions now.