Here's an example with short exact sequences of (infinitely generated) abelian groups.

Fix a prime $p$ and let $C=\mathbf{Z}[1/p]/\mathbf{Z}$ be the $p$-Prüfer quasi-cyclic group; let $C^{(n)}$ be an isomorphic copy of $C$, and let $C^{(n)}_k\simeq\mathbf{Z}/p^k\mathbf{Z}$ be the $p^k$-torsion in $C^{(n)}$. 

Consider $B=\bigoplus_{n\in\mathbf{N}}C^{(n)}$. Let $\mathbf{N}^*$ denote the positive integers. Define subgroups $A=\bigoplus_{n\in\mathbf{N}^*}C^{(n)}_{n}$ and $A_2=\bigoplus_{n\in\mathbf{N}}C^{(2n)}_{n}$. Then clearly, $A$ is isomorphic to $A_2$, and $B/A$ and $B/A_2$ are isomorphic (to $B$). On the other hand, the exact sequences $A\hookrightarrow B\twoheadrightarrow B/A$ and $A_2\hookrightarrow B\twoheadrightarrow B/A_2$ are not isomorphic, because $A$ contains the $p$-torsion of $B$ while $A_2$ does not.