Finite abelian group is isomorphic to its Pontryagin dual

Statement
Suppose $$G$$ is a finite abelian group and $$\hat{G}$$ is its Pontryagin dual. Then, $$\hat{G}$$ is also a finite abelian group and $$G \cong \hat{G}$$. However, there is no canonical isomorphism between the groups, and groups acting on $$G$$ may have corresponding actions on $$\hat{G}$$ with very different orbit structures.