TITLE: Universal group?
QUESTION [24 upvotes]: 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.

REPLY [5 votes]: Аnswered to move the question to answered status.
Thank you all for your comments they were helpful for me and Dima.