Examples of non-isomorphic abelian groups which are part of exact sequences Suppose $A_1$, $A_2$, $A_3$ and $B_1$, $B_2$, and $B_3$ are two
short exact sequences of abelian groups.
I am looking for two such short sequences where $A_1$ and  $B_1$ is isomorphic
and $A_2$ and $B_2$ are isomorphic but $A_3$ and  $B_3$ are not.
(Similarly I would like examples in which two of the other pairs are isomorphic 
but the third pair are not, etc)
 A: For the first pair take 
$$0 \longrightarrow \mathbb{Z} \stackrel{2}{\longrightarrow} \mathbb{Z} \longrightarrow \mathbb{Z} / 2\mathbb{Z} \longrightarrow 0$$
and
$$0 \longrightarrow \mathbb{Z} \stackrel{3}{\longrightarrow} \mathbb{Z} \longrightarrow \mathbb{Z} / 3\mathbb{Z} \longrightarrow 0.$$
For sequences with non-isomorphic first pairs you can use an infinite direct sum of $\mathbb{Z}$'s and include one or two copies of $\mathbb{Z}$. The quotient will be the infinite direct sum again so the second and third pairs are isomorphic but the first pair will be non-isomorphic.
Finally for non isomorphic central pairs take
$$0 \longrightarrow \mathbb{Z} / 2\mathbb{Z} \longrightarrow \mathbb{Z} / 2\mathbb{Z} \times \mathbb{Z} / 2\mathbb{Z} \longrightarrow \mathbb{Z} / 2\mathbb{Z} \longrightarrow 0$$
and
$$0 \longrightarrow \mathbb{Z} / 2\mathbb{Z} \stackrel{2}{\longrightarrow} \mathbb{Z} / 4\mathbb{Z} \longrightarrow \mathbb{Z} / 2\mathbb{Z} \longrightarrow 0.$$
A: Here are some finite examples:
$$\begin{array}{c} 0 \to& C_2\times C_2 &\to& C_2\times C_2\times C_4 &\to& C_4 &\to 0 \\
 0 \to& C_2\times C_2 &\to& C_2\times C_2 \times C_4 &\to& C_2\times C_2 &\to 0 \\
 0 \to& C_4 &\to& C_2\times C_2 \times C_4 &\to& C_2 \times C_2 &\to 0 \\
 0 \to& C_4 &\to& C_2\times C_8 &\to& C_2 \times C_2 &\to 0 \end{array}$$
