Determine all abelian group $G$ such that $0 \to \mathbb{Z} \to G \to \mathbb{Z}_2^k \to 0$ is exact

Question

I came across the following exact sequence of abelian group

$0 \to \mathbb{Z} \to G \to \mathbb{Z}_2^k \to 0$

and I want to find all possible $G$. I know that the answer would be "use Ext", but I don't know what it is. Using the formal rules I can conclude that $\text{Ext}(\mathbb{Z}, \mathbb{Z}_2^k) \cong \text{Ext}(\mathbb{Z}, \mathbb{Z}_2)^k \cong \mathbb{Z}_2^k$, i.e. there would be $2^k$ extensions.

I can find only $G \cong \mathbb{Z}\oplus \mathbb{Z}_2^k$ or $G \cong \mathbb{Z}\oplus \mathbb{Z}_2^{k-1}$. Which are the others $2^k - 2$?