Let $G$ be an abelian group and let $A$ and $B$ be subgroups of $G$. Furthermore, let $C$ be a subgroup of $A \cap B$. I would like to find another subgroup $A+B \subseteq D \subseteq G$ so that $D/(A+B) \cong (A \cap B)/C$. In general I'm not sure this is possible although in the situation I am really interested in I am pretty sure it is true. Therefore, my questions are:
1) Are there additional restrictions that I can place on the situation to guarantee the existence of such a subgroup $D$?
2) If a $D$ exists and I know a set of representatives for $(A \cap B)/C$ is it possible to deduce a set of representatives for $D/(A+B)$.