Two abelian groups with the same order are isomorphic?

Question

True of false: if G and H are two groups with the same order and both are abelian, then they are isomorphic.

Answer 1

Only if the order is square-free.

Answer 2

False.

$\mathbb{Z_4}$ and $\mathbb{Z_2}\times\mathbb{Z_2}$ have order $4$ but not isomorphic