Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the normalizer $N_G(H)$ of $H$.

Now let $H$ be an arbitrary reductive subgroup of $G$. Is it true that $Z_G(H)\cdot H$ is of finite index in $N_G(H)$?

In the case of certain Lie groups, the answer is yes, see e.g. D. Poguntke, Normalizers and centralizers of reductive subgroups of almost connected Lie groups (MR1650341), which motivates my question.