Finding all abelian groups such that there exists certain short exact sequence. I have to find all abelian groups $A$ such that there exists a short exact sequence $0\rightarrow\mathbb{Z}\rightarrow A\rightarrow\mathbb{Z}\oplus\mathbb{Z}_{6}\rightarrow 0$.
I have found $A=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}_{6}$, $A=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}_{2}$, $A=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}_{3}$, and $A=\mathbb{Z}\oplus\mathbb{Z}$. I am not able to find any other, but I am also not able to prove these are the only ones.
This exercise has come up in an Algebraic Topology course, so I would like to prove it using elementary abstract algebra and not Homological Algebra.
Any hint will be greatly appreciated.

Hint: For group extensions in a short exact sequence 
$$0\to A\to E\to B\to 0$$
for each of the homomorphisms $B\to{\rm Out}A$ there is an extension $E$.
Also ${\rm Out}\ \Bbb{Z}={\Bbb{Z}}_2$.
