I ran across this question on an old University of Maryland Algebra qual (1993, 4a):
Consider an exact sequence of abelian groups $0\rightarrow \mathbb{Q} \rightarrow B \rightarrow \mathbb{Z}_6\rightarrow 0$. Show that $B\cong \mathbb{Q}\oplus \mathbb{Z}_6$.
I know that there are examples of exact sequences of abelian groups which do not split, such as $0\rightarrow \mathbb{Z}_2 \rightarrow \mathbb{Z}_4 \rightarrow \mathbb{Z}_2 \rightarrow 0$, so my guess is that the presence of $\mathbb{Q}$ plays a role in the existence of a back-map from $B$, although I've had no luck thus far in constructing it.