Group of abelian extensions is quotient of group of symmetric 2-cocycles by group of 2-coboundaries for trivial group action

Statement
Suppose $$A$$ and $$G$$ are abelian groups. Then, the set of abelian groups $$E$$ with $$A$$ identified as a normal subgroup of $$E$$, and an isomorphism $$E/A \to G$$, can be identified with the elements of the quotient group of the group of symmetric 2-cocycles for the trivial group action of $$G$$ on $$A$$ by the group of coboundaries for the trivial group action.

This group can also be identified with the group $$\operatorname{Ext}^1(G,A)$$, denoted for short as $$\operatorname{Ext}(G,A)$$.