Normal subgroup of the identity component of a linear Lie group is normal in the whole group?

Question

Suppose $G$ is a linear Lie group (i.e. $G$ admits a finite dimensional faithful representation) and $G$ has finitely many connected components. Let $G_0$ be the identity component of $G$. If $N$ is a normal subgroup of $G_0$, is $N$ necessarily normal in $G$?

Answer 1

No. Take $H$ to be a connected group, and $G=(\prod _{i=1}^n H)\rtimes S_n$. Where $S_n$ acts by permuting the factors. Then $G^0=\prod _{i-1}^n H$, and each of the factors $H$ is a normal subgroup in $G^0$ which is not normal in $G$.