Given a map $f: S \rightarrow H$, with $S$ some set of generators, when and how can $f$ be extended to a well defined homomorphism with $\phi : G \rightarrow H $ with $G$ a group generated by $S$ with $\lbrace r_i \rbrace$ a finite set of relations.
Further what requirements are needed to show that $\phi[G] = H$ and that $\phi$ is an isomorphism.
$G$ is the quotient of $F(S)$ the free group generated by $S$ by its normal subgroup $N$ generated by the relation. $f$ induces a morphism $g:F(S)\rightarrow H$, it factors by $G$ if and only if its kernel contains $N$. This is equivalent to saying that $g(r_i)=1$.