Algebraic group extension

Definition
Suppose $$A$$ and $$B$$ are algebraic groups.

An algebraic group extension with normal subgroup $$A$$ and quotient group $$B$$ is defined as a group $$G$$ with a specified closed normal subgroup $$N$$ having a specified algebraic group isomorphism to $$A$$ and a specified algebraic group isomorphism from the quotient group $$G/N$$ to $$B$$.

In group theory, such a $$G$$ is termed an extension of $$A$$ (the subgroup isomorphic to the normal subgroup) by $$B$$ (the subgroup isomorphic to the quotient group). In some other areas of mathematics, particularly geometric group theory, $$G$$ is termed an extension of the quotient by the normal subgroup.

The algebraic group extension problem seeks to classify all group extensions with a specified closed normal subgroup and a specified quotient group.