Suppose the we have an epimorphism $s\colon M\to N,$ where $M$ is a free $R$-module of rank $r$ and $N$ is a finitely generated $R$-module, such that there exists a basis $B:=\{m_{1},\dots, m_{r}\}$ of $M$ with $s(m_{i})\neq0\in N$ for all $i\in\{1,\dots,r\}.$ Let $N$ be generated by $0\notin\{n_{1}, \dots, n_{r}\}.$ What is the weakest hypothesis that I have to assume so I can affirm that there exist a basis $B'=\{e_{1},\dots,e_{r}\}$ of $M$ with $e_{i}\in s^{-1}(n_{i})$ for all $i\in\{1,\dots,r\}$? In what interesting cases I can say this?

I am particularly (but not only) interested in the case where $s$ is a quotient map $s\colon M:=\mathbb{R}[x_{0},\dots,x_{n}]_{\deg_{x_{0}<r}}\to N:=(\mathbb{R}[x_{0},\dots,x_{n}]/(h))_{\deg_{x_{1},\dots, x_{n}}>e},$ for some integer $e,$ with $h\in\mathbb{R}[x_{0},\dots,x_{n}]$ an homogeneous polynomial of degree $d<r$ with leading term $x_{0}^{d}$ in the homogenizing variable $x_{0}$ and we consider these spaces as $\mathbb{R}[x_{1},\dots,x_{n}]$-modules. I have a generating set of $N$ as described and want to pull it back to a basis of $M.$