Let $G$ be *simply connected, simple* algebraic group over $\mathbb{C}$.
Let $H\subset G$ be a *self-normalizing* spherical subgroup of $G$,
not necessarily connected or reductive.
Here "self-normalizing" means that $\mathcal{N}_G(H)=H$.

I am interested in a certain quotient $H^{\mathrm{mt}}$ of $H$. Namely, let $H^0$ denote the identity component of $H$, and let $H^{\mathrm{u}}$ denote the unipotent radical of $H^0$. Set $H^{\mathrm{red}}=H^0/H^{\mathrm{u}}$, which is a connected reductive group. Let $H^{\mathrm{ss}}$ denote the commutator subgroup of $G^{\mathrm{red}}$, which is a connected semisimple group. Set $H^{\mathrm{tor}}=H^{\mathrm{red}}/H^{\mathrm{ss}}$, which is a torus. Then $H^{\mathrm{u}}$ is a normal subgroup in $H$, and $H^{\mathrm{ss}}$ is a normal subgroup in $H/H^{\mathrm{u}}$. We set $$ H^{\mathrm{mt}}=(H/H^{\mathrm{u}})/H^{\mathrm{ss}}.$$ Then we have a short exact sequence $$ 1\to H^{\mathrm{tor}}\to H^{\mathrm{mt}}\to\pi_0(H)\to 1,$$ where $\pi_0(H)=H/H^0$.

Question 1.Is it true that the finite group $\pi_0(H)$ is always abelian?

Question 2.When $\pi_0(H)$ is abelian, is it true that it acts on $H^{\mathrm{tor}}$ trivially and that the group $H^{\mathrm{mt}}$ is abelian (hence of multiplicative type)?

Question 3.What are nontrivial examples of $H$ and $H^{\mathrm{mt}}$ (say, when $\pi_0(H)$ is not cyclic, or when $\mathrm{dim}(H^{\mathrm{tor}})>1$, but $H$ is not parabolic, etc.)?

Motivation: I would like to compute the Tate-Shafarevich kernel $Ш^2(k,H)$ (whatever this means) over a number field $k$, and then $H^{\mathrm{u}}$ and $H^{\mathrm{ss}}$ play no role, so I consider the quotient $H^{\mathrm{mt}}$ of $H$.

Any examples and/or references will be appreciated.