Let $g \geq 2$. Let $S = \langle a\_1,b\_2,...,a\_g,b\_g | [a\_1,b\_1] \cdots [a\_g,b\_g] \rangle$ be the fundamental group of a genus $g$ surface and let $F\_g$ be a free group with $g$ generators. Given two surjections $f\_1,f\_2 : S \to F\_g$ is there a way to determine if there are automophisms $\phi: S \to S$ and $\psi: F\_g \to F\_g$ so that $f\_1 = \phi \circ f\_2 \circ \psi$? 


Is there an example of two surjections $f\_1,f\_2$ that are not equivalent in the above way? 


I asked the question on [MSE](https://math.stackexchange.com/questions/2700834/equivalance-of-surjections-from-a-surface-group-to-a-free-group) before but didn't get much.



Equivalence of surjections from a surface group to a free group
This is true, and it is written up in lemma 2.2 of "The co-rank conjecture for 3--manifold groups" by C. Leininger and A. Reid [<https://arxiv.org/abs/math/0202261>](https://arxiv.org/abs/math/0202261). They state the result in slightly different language, that is they prove that any such epimorphism is induced by choosing a genus $g$ handlebody. 


