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.
• My 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).