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](http://www.heldermann-verlag.de/jlt/jlt08/POGULAT.PDF) ([MR1650341](http://www.ams.org/mathscinet-getitem?mr=1650341)), which motivates my question.