Algebraic second cohomology group

Definition
Let $$G$$ be an algebraic group and $$A$$ be an abelian algebraic group, both over a field $$K$$, with an action $$\varphi:G \to \operatorname{Aut}(A)$$ that is a regular function viewed as a map $$G \times A \to A$$.

Definition in cohomology terms
The algebraic second cohomology group $$H^2_{\varphi,\mbox{alg}}(G,A)$$ is an abelian group defined in the following equivalent ways.

When $$\varphi$$ is understood from context, the subscript $${}_\varphi$$ may be omitted in the notation for the cohomology group, as well as the notation for the groups of 2-cocycles and 2-coboundaries. Also, the subscript $${}_{\mbox{alg}}$$ may be omitted in contexts where it is understood that we are interested in algebraic cohomology only.

NOTE: Apart from Definition (1), all definitions need to be fixed to be made "algebraic."

Definition in terms of group extensions
There is an alternative definition of $$H^2_{\varphi,\mbox{alg}}(G,A)$$ that is specific to 2 and has no easy analogue for other $$H^n_{\varphi,\mbox{alg}}(G,A)$$. This is in terms of group extensions.

$$H^2_{\varphi,\mbox{alg}}(G,A)$$ can also be identified with the set of congruence classes of algebraic group extensions with closed normal subgroup isomorphic to $$A$$ and quotient group isomorphic to $$G$$ where the induced action of the quotient is the specified action $$\varphi$$. By a group extension, we mean a group $$E$$ having $$A$$ as a normal subgroup and $$G$$ as a quotient group. Two extensions $$E_1$$ and $$E_2$$ are congruent if there is an algebraic group isomorphism of $$E_1$$ to $$E_2$$ which is identity on $$A$$ and induces the identity map on $$G$$ as a quotient.