Is a normal subgroup normal in a normal group?

Question

Let $G$ be a normal ($T_4$) topological group, that is, every two disjoint closed sets of $G$ have disjoint open neighborhoods. Let $H$ be a subgroup of $G$ that is normal too ($T_4$) with the induced subspace topology. Is $H$ a normal subgroup of $G$? That is, $\forall g \in H \, $, $ \, gH=Hg \ \ $?

This question came up as a pun and evolved into a sincere question to which I did not get a definitive answer.

Answer 1

No, take any group $G$ with a non-normal subgroup $H$ and equip $G$ with the discrete topology.

Answer 2

Not necessarily. Take any group $G$ endowed with the discrete metric and any non-normal (in the group-theoretical sense) subgroup $H$ of $G$. Then $H$ is normal with respect to the induced topology, but it is still a non-normal subgroup of $G$.