$\begin{array}{ccccccccc} &&0&&0&&0\\ &&\downarrow &&\downarrow && \downarrow\\ 0 & \to & \mathbb{Z}_2\{a\} & \to & \mathbb{Z}_2\{a\} & \to & 0 & \to & 0\\ & &\downarrow & & \downarrow &&\downarrow\\ 0&\to&\mathbb{Z}_2\{a\}\oplus\mathbb{Z}_2\{b\} & \xrightarrow{f} & G & \xrightarrow{g} & \mathbb{Z}_2\{c\} & \to & 0\\ &&\downarrow & &\downarrow & & \downarrow\\ 0 & \to & \mathbb{Z}_2\{b\} & \xrightarrow{h} & \mathbb{Z}_2\{y\}\oplus\mathbb{Z}_2\{z\} & \xrightarrow{i} & \mathbb{Z}_2\{c\} & \to &0\\ &&\downarrow && \downarrow &&\downarrow\\ &&0 && 0 && 0 \end{array}$
where $\mathbb{Z}_2\{a\}$ means that $a$ generates $\mathbb{Z}_2$ and $y=h(b)$ and $i(z)=c$.
In the diagram, first and third rows are split and first and third columns are split.
Then second row($0 \to \mathbb{Z}_2 \oplus \mathbb{Z}_2 \to G \to \mathbb{Z}_2 \to 0$) or second column($0 \to \mathbb{Z}_2 \to G\to \mathbb{Z}_2 \oplus \mathbb{Z}_2 \to 0$) is split? Or $G$ is isomorphic to $\mathbb{Z}_4 \oplus \mathbb{Z}_2$?