Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group

Statement
Suppose $$G$$ is a fact about::finite abelian group (i.e., it is both a finite group and an fact about::abelian group). Let $$C(G)$$ be the set of conjugacy classes of $$G$$ and $$R(G)$$ be the set of irreducible representations of $$G$$ over $$\mathbb{C}$$. The automorphism group $$\operatorname{Aut}(G)$$ acts on the sets $$C(G)$$ and $$R(G)$$. The claim is that the orbit sizes in both sets under the action of $$\operatorname{Aut}(G)$$ are the same. In other words, there is a size-preserving bijection between the sets of orbits in $$C(G)$$ and $$R(G)$$.

Related facts

 * Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group
 * Number of orbits of irreducible representations need not equal number of orbits of conjugacy classes under automorphism group