Splitting of free groups

Question

Let's assume there are two epiomorphisms (with non-trivial kernels) $f_1,f_2$ from nonabelian finitely-generated free group $f_i:F \rightarrow G$ into arbitrary finitely generated group. From the property of free groups I can lift $f_1$ to $g:F \rightarrow F$, which gives me a commuting triangle. In what situation can I say that there exists a finite index subgroup $H <F$ with an isomorphism $\varphi: T * I \rightarrow H$ such that $g \circ \varphi $ takes $I$ isomorphically to an image of $g$ ? Thank you for all your answers