On decomposition of subgroups a finite abelian groups

Question

Let $G$ be an finite abelian group and $H$ is a subgroup of $G$. Then do can find a decomposition of $G$ as a direct sum of cyclic groups such that the intersections of the summands with $H$ give a direct sum decomposition of $H$?

Answer 1

I am sorry that I previously wrote an incorrect proof of this claim. As you have discovered yourself, the answer is no.

A counterexample is $G = \langle a \rangle \times \langle b \rangle$ with $|a|=8$, $|b|=2$ and $H = \langle a^2b \rangle$. If the claim were true, then $H$ would be contained in a cyclic subgroup of order $8$, but this is not true because $a^2b$ is not the square of any element of $G$.