Suppose I have a group $G$, which I know is finitely presentable and infinite. (In particular, I have a presentation for it, though not the one I want).
Suppose I have a small list of generators $\{g_1,\ldots,g_n\}$ for $G$.
Is there an algorithm to find a presentation for $G$ using the given generators?
More specifically, I have an exact sequence of finitely presented groups $$1\rightarrow F_2\rightarrow G\rightarrow H\rightarrow 1$$ Here $F_2$ is the free group on 2 generators, and I have presentations for both $G$ and $H$. I'd like to find a presentation of $G$ where the generators have the form $\{x,y,g_1,\ldots,g_k\}$, where $x,y$ are the two free generators of $F_2$.
It looks like GAP would allow me to do this for finite groups $G$, though the same function does not appear to work for infinite groups.