Let $G_1$ and $G_2$ be finitely presentable groups and let $f : G_1 \rightarrow G_2$ be a surjective homomorphism. Denoting the kth term of the lower central series of $G_i$ by $\gamma_k(G_i)$, assume that $f$ induces an isomorphism $G_1 / \gamma_k(G_1) \rightarrow G_2 / \gamma_k(G_2)$ for all $k \geq 1$. Is $f$ necessarily an isomorphism?

EDIT : I forgot the obvious assumption that the intersection of the lower central series of $G_i$ is trivial for $i=1,2$.