I'm just trying to solve an exercise from "A course in homological algebra" by Hilton and Stammbach, however I couldn't find any example.
Find two short exact sequences of abelian groups
$0 \longrightarrow A'\longrightarrow A \longrightarrow A'' \longrightarrow 0$
$0 \longrightarrow B'\longrightarrow B \longrightarrow B'' \longrightarrow 0$
such that two of the abelian groups belonging to distinct sequences are isomorphic, however the third is not, e.g., $A \cong B$ and $A'\cong B'$, but $A''\ncong B''$.
I have already found an example using semi direct product, however the resulting group is not abelian, just pick $A = G\ltimes H$ and $B = G \bigoplus H$. I think that two subgroups $HK = HN$ of a group $G$ such that $K \neq N$ will suffice one of the cases, however I did not found any concrete example.
Thanks in advance.