Fix a group $G$ and fix a presentation of $G$ as $\langle X\mid R\rangle$. A *presentationally finite* extension of $G$ is any group that can be presented as $H=\langle X\cup X'\mid R\cup R'\rangle$, where $X',R'$ are finite. (Mind, the natural homomorphism $G\to H$ may not be injective. If necessary, we can restrict ourselves to cases where it is.)


For example, the presentationally finite extensions of the trivial group are the finitely presentable groups.



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> **Question**: Is there an existing name for this kind of group homomorphism/extension?
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Unless I've missed something, it's not very hard to see that this does not depend on the choice of presentation of $G$ (sketch: adding extraneous generators to $X$ and the corresponding generation relations to $R$ does not change the isomorphism type (over $G$) of $\langle X\cup X'\mid R\cup R'\rangle$, neither does adding any relations that follow from the existing ones; thus, we can just add all elements of $G$ plus any number of their copies we want, and all relations true in $G$ about them and then remove any redundant ones we do not want to obtain all possible presentations).


Thus, this is really a property of the map $G\to H$ (or the extension $G\leq H$ if the map is injective) and not of a particular presentation.


I'm mostly interested in the case of finitely generated groups, but I think the idea works just as well for arbitrary groups, of arbitrary cardinality.


(Another way to describe this is simply as the groups of the form $H=(G\*F)/N$, where $F$ is a free group of finite rank and $N$ is finitely generated as a normal subgroup of $G\*F$, such that $N\cap G$ is trivial if we demand that the map $G\to H$ actually be injective.)



Presentationally finite group "extensions"
The kind of presentation you are giving is called a relative presentation and has been studied extensively by Steve Pride and his coauthors. So relatively finitely presented over $G$ is probably the best name (I didn't search his papers to see if he uses this term.)


