Let $F$ be a finitely generated free group. Consider $R\subset F$ a finite set of relations and denote with $G$ the quotient group of $F$ by the normal closure of $R$.

Now suppose we are given another set of generator $g_1,\ldots,g_k$ of $G$.

The question is the following: is it possible to find $y_1,\ldots,y_k\in F$ such that

the image of $y_i$ in G is $g_i$ for every $1\leq i\leq k$,

the set $\{y_1,\ldots,y_k,R\}$ generates $F$ ?

If it is not the case, is it possible if we suppose that $G$ is free?

I'm able to answer the question only in one case: if $R=\{r\}$ and $G$ is free, then $r$ is an element of a base of $F$ so the answer is yes. I learned the proof of this fact in Lyndon-Schupp's book and it uses a kind of Freiheitssatz, so it doesn't help too much for the general case.