Assume that finitely generated groups $G$ and $H$, are not finitely presented. Fix a generating set $\mathfrak g:=\{g_1,\dotsc,g_n\}$ of $G$. Let $\mathfrak R:=\{R_1,R_2,\dots\}$ be the set of all relations in $G$. For all nonempty finite subsets $\mathfrak S$ of $\mathfrak R$, there exists a generating set of $H$ whith same size as $\mathfrak g$, which its corresponded set of relations, contain $\mathfrak S$. What we can say about $G$ and $H$. Are they isomorphic? Are they quotient of each other?
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$\begingroup$ In what set does $\mathfrak R$ live? Is it another name for $\ker(\operatorname{Free}(\mathfrak g) \to G)$? $\endgroup$ – LSpice Jan 20 '16 at 18:32

$\begingroup$ @LSpice: Yes, it is exactly all of $\ker(\text{Free}(\mathfrak g)\rightarrow G)$. $\endgroup$ – Meisam Soleimani Malekan Jan 20 '16 at 19:06

$\begingroup$ You may like to look up the GromovGrigorchuk space of marked groups. If I've understood you correctly, your condition means that $G$ is in the closure of the subspace of groups isomorphic to $H$. $\endgroup$ – HJRW Jan 20 '16 at 22:44

1$\begingroup$ @YCor, it seems to me that the OP would benefit from an example of a group G which is a limit of groups H, but such that H is not a quotient of G. I'm under the impression that such a (necessarily infinitely presented) G should exist, but don't know a construction. Do you know one? $\endgroup$ – HJRW Jan 21 '16 at 12:06

1$\begingroup$ @YCor, I agree that what you describe is a contradiction to the question as literally read. But I presume the OP is looking for some nontrivial phenomenon in this vicinity, and so I thought an example of that form might be helpful to him. It was just a suggestion. $\endgroup$ – HJRW Jan 21 '16 at 19:58