If $H \le K \le G$ then $H \trianglelefteq \overline{K}$?

Question

If I have topological groups $H \le K \le G$, I could prove that if $H \trianglelefteq K$ then $\overline{H} \trianglelefteq K$ but is it also true that $H \trianglelefteq \overline{K}$?

Answer 1

If $H$ is a normal subgroup of $G$, then it is normal in any intermediate subgroup $K'$, since it is closed under conjugation by any element $g\in K'$.
In particular, this holds for $K'=\bar K$.

So, the following statements are true in this case: $$H\trianglelefteq K,\quad H\trianglelefteq \bar K,\quad \bar H\trianglelefteq \bar K\,.$$ However, the forth statement, $\bar H\trianglelefteq K$ only holds if $\bar H\subseteq K$, which is not guaranteed in general, for example if $K$ is assumed to be closed.