The first example that comes to mind, $G=\bigoplus_{i=1}^\infty\mathbb{Q}$ and $H=\mathbb{Q}/\mathbb{Z}\oplus\bigoplus_{i=1}^\infty\mathbb{Q}$ seems to work.

The first example that comes to mind, $G=\bigoplus_{i=1}^\infty\mathbb{Q}$ and $H=\mathbb{Q}/\mathbb{Z}\oplus\bigoplus_{i=1}^\infty\mathbb{Q}$, seems to work.

FYI, a related example $G=\bigoplus_{i=1}^\infty\mathbb{Q}$ and $H=\mathbb{Z}\oplus\bigoplus_{i=1}^\infty\mathbb{Q}$ handles your question with "surjective" replaced by "injective".