Let $G$ be an abelian group, and $H$, $N$ are subgroup of G. Assume that $$ G/H \cong N $$ and $N$ is flat as a $\mathbb{Z}$-module. Then, are $H$ and $N$ direct summand of G? Is there a counter example? I know if $N$ is projective as a $\mathbb{Z}$-module, $G \cong H \oplus N$.
Thank you for your directions!