Consider a finitely generated group $G$ and a subnormal series of $G$: $$1=G_0\trianglelefteq G_1\trianglelefteq\cdots\trianglelefteq G_{n-1}\trianglelefteq G_n=G$$ Now, suppose that $G_1$ fibers, i.e. there is an epimorphism $\varphi:G_1\to\mathbb{Z}$ with $\ker(\varphi)$ finitely generated. Is there any way to lift the fibration onto $G$? i.e. to construct an epimorphism $\varphi':G\to\mathbb{Z}$ with $\ker(\varphi)$ finitely generated.

I know the answer is 'no' in general, but maybe it is true if we assume some extra properties such as $G_i/G_{i-1}$ being $F_\infty$. More generaly, I would like to study this problem when we know that $G_i/G_{i-1}$ fibers insted of the first quotient.