This is probably elementary but I know that in group theory the devil is in the details so I want to check. Assume I've a finitely generated group $G$ and a normal subgroup $U$ (if it does matter, in my situation $G$ is a semidirect product of $U$ by some $H$). Let $g_0,\dots,g_n$ be the generators of $G$. Assume that I know a presentation of $U$ and that $g_0 \not \in U$ (or even that $g_0 \in H$). Then, I want to find a presentation of the image of $U$ in the quotient $G/\langle g_0\rangle$.
Formally what I'm looking for is the intersection between the normal closure of $g_0$ and $U$ but it's hard to make it explicit. On the other hand, it seems to me that the only way taking the quotient can add relations is as follows: since $U$ is normal, $g_0$ acts on it by conjugation, so for $u\in U$, let $w_u$ be $g_0ug_0^{-1}$ written as a word in the generators of $U$. Then, taking the quotient by $g_0$ forces this action to be trivial, hence add the relation $w_u\equiv u$
Indeed I believe that there are no other new relations but didin't manage to write down a rigorous proof.