Two groups $G_1,G_2$ are (abstractly) commensurable if there exist isomorphic finite-index subgroups $H_1 \leq G_1$, $H_2 \leq G_2$.
Question: If $G_1,G_2$ are commensurable, do there exist two isomorphic normal finite-index subgroups $H_1 \lhd G_1$, $H_2 \lhd G_2$?
Here, I am assuming that $G_1$ and $G_2$ are finitely generated. As a consequence, one of the two subgroups can be chosen normal, but can they be chosen normal simultaneously? If the answer is negative in full generality, I am also interested in partial positive answer. For instance, we may assume that $G_1$ and $G_2$ are residually finite.