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 before but didn't get much.

veryconfused. $\endgroup$ – PseudoNeo Oct 15 '18 at 2:03