This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup.

Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic $p>0$. Let $H\subset G$ be a $k$-subgroup, not necessarily smooth. Question 1: Does the quotient $G/H$ exist as a $k$-variety?

I am interested in the following special case. Let $H^{\rm mult}$ denote the largest quotient of $H$ which is a $k$-group ($k$-group scheme) of multiplicative type. Set $H_1=\ker[H\to H^{\rm mult}]$. I assume that $H_1$ is smooth, connected and semisimple. Question 2: Does the quotient $G/H$ exist as a $k$-variety under this assumption? (I do not assume that $H^{\rm mult}$ is smooth.)

All comments and references are welcome!