This is a continuation of my question http://mathoverflow.net/questions/16261/.

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!