I have a question on quotients of linear algebraic groups. Let $G$ be a linear algebraic group and $H$ a linear algebraic group acting on $G$ as an algebraic group.
I would like to know what the quotient $G/H$ is, in several categories (i.e. we have the forgetful functor from linear algebraic groups to (including but not only) $(Sets)$, $(Groups)$, $(affine-schemes)$, etc.)
My question is: in which of those categories does such a quotient exist (under which assumptions) and when does taking the quotient commute with the several forgetful functors? Further, is any of that easy to see?
I know this question is quite open (and some things are also clear, e.g. for the quotient to exist in Groups we should ask $H$ to act as a normal divisor of $G$), but I'd be happy about any insight.