I think your work was heading in the right direction. I will try to formalize what you were saying below.
Let be the group multiplication. G/A acts on A in the following way: Let Then the action is defined by Note that the definition of is legitimate, because is normal in so The tough work is showing that is well-defined. That is well-defined follows from the fact that is abelian.
Does this help clear things up? Let me know if anything is unclear. Good luck!