Let $S$ be a fixed base scheme and $G, H$ be group schemes over $S$. Since I am mainly interested in commutative group schemes over fields, we may assume that $G,H$ are commutative and $S$ is a field if this helps.

(1) Let $f:G\to H$ be a morphism of group schemes. To define the cokernel of this map, we need to choose which topology to work with. Some people use the fppf topology (as in van der Geer & Moonen's book) and other people use the fpqc topology (as in Cornell-Silverman). My question is: what is the difference of those two topologies in terms of group schemes? Is fppf quotient and fpqc quotient of group schemes different? Which topology do people prefer when they are working with group schemes?

(2) Let $H$ be a (normal) closed subgroup scheme of $G$. I think there are at least three plausible definitions of the quotient $G/H$:

Categorical quotient: Since $H$ naturally acts on $G$, we can think categorical quotient $G/H$ of the action $H\times G\to G$.

Fppf/fpqc quotient: $G/H$ represents the quotient of $H\to G$ in the category of fppf/fpqc sheaves.

Naive quotient: A group scheme $G/H$ with a surjective (wrt fppf/fpqc topology) map $p:G\to G/H$ such that kernel of $p$ is the inclusion $H\to G$

Are they equivalent in some good situations? In van der Geer & Moonen's book, it is proved that a fppf quotient is also a categorical quotient. But I cannot find proof nor prove other directions.

context of the question (2): Let $f:A\to B$ be an isogeny of abelian varieties with kernel $\ker f$. Then we have the dual exact sequence $0\to \widehat{B}\to \widehat{A}\to \widehat{\ker f}\to 0$. In Milne's book on abelian variety, to prove the dual exact sequence, consider $0\to \ker f\to A\to B\to 0$ as an exact sequence in the category of commutative group schemes over a field and use a long exact sequence with $\text{Hom}(-, \mathbb{G}_m)$. To use the long exact sequence, we need to prove $B$ is $A/\ker f$ as a fppf/fpqc quotient (In fact I don't know which topology to work with. This is why I ask the question (1)...). However, I only know that $B$ is the `naive quotient (3)' $A/\ker f$.

(3) Is the category of commutative group schemes over a field an abelian category? This statement is in Milne's book on abelian variety, but I cannot find proof. The main point is existence of cokernel, i.e. representability of fppf/fpqc quotient. However, I only know the following theorem in Cornell & Silverman,

**Theorem**. Let $G$ be a finite type $S$-group scheme and let $H$ be a closed subgroup scheme of $G$. If $H$ is proper and flat over $S$ and if $G$ is quasi-projective over $S$, then the quotient sheaf $G/H$ is representable.

and this is too weak to prove our statement.

Also one more quick question: do you know any good reference dealing with sufficiently general group schemes? I know Shatz's paper in Cornell-Silverman, Tate's paper in Cornell-Silvermann-Stevens, and Stix's lecture note, but they focus on finite flat group schemes. Also, I know some other articles & books which mainly focus on affine algebraic groups. Are there some more general references?

Thank you for reading my stupid questions.