Isomorphism between tensor products of abelian groups

Question

Let $A,A',B,C,D,D'$ be (finitely generated free) abelian groups, s.t. $B \subseteq A' \subseteq A$, $D \subseteq D' \subseteq C$ and $D' \subseteq A'$. All quotients are finite abelian groups. Let $A/A' \cong D'/D$. Does $A/B \otimes_{\mathbb{Z}} C/D \cong A'/B \otimes_{\mathbb{Z}} C/D'$ hold?

EDIT: Before, I mistakenly assumed $A/D \cong A'/D'$ instead of $A/A' \cong D'/D$.

Answer 1

Let $A=C=\mathbb{Z}$, $A'=B=D'=2\mathbb{Z}$, and $D=4\mathbb{Z}$. Then $A/A'\cong D'/D$, but $A/B \otimes C/D\cong \mathbb{Z}/2$ while $A'/B \otimes_{\mathbb{Z}} C/D'\cong 0$.