Suppose you have an exact sequence $1\to G_1\to G_2\to G_3\to 1$ of affine group schemes, over a field $k$. By this a mean that $G_2\to G_3$ is a quotient map (i.e., the map on coordinate algebras is injective, or faithfully flat), and the kernel is $G_1$. Moreover, assume $G_1$ is commutative (if necessary).
Is it true that the exact sequence is the pull-back of an exact sequence of the form $1\to G_1\to G_2'\to G_3'\to 1$, where $G_2'$ and $G_3'$ are algebraic quotients of $G_2$ and $G_3$ respectively?
On a related question: if $\overline k$ is an algebraic closure of $k$, then $Gal(\overline k/k)$ acts on each $G_i(\overline k)$ but not necessarily in a continous way if the $G_i$ are not algebraic. Do you still have some sort of Galois cohomology exact sequence?