Finite index dense normal subgroups of completely metrizable groups

Question

Is there some completely metrizable group $M$, which contains a normal subgroup $N\trianglelefteq M$ of finite index (at least $2$) that is dense in $M$?

Answer 1

Let $\Bbb F_p$ be the field with $p$ elements, considered as a discrete group and consider the group $\prod_{n\in\Bbb N}\Bbb F_p$, which is Polish.

The kernel of a discontinuous linear functional $\prod_{n\in\Bbb N}\Bbb F_p\to\Bbb F_p$ will be a dense subgroup of index $p$.