Am I doing it wrong or is this an exception to the split rules?
There is a short exact sequence $$1 \longrightarrow \mathbb Z/_{2\mathbb Z} \hookrightarrow \mathbb Z/_{4\mathbb Z} \twoheadrightarrow \mathbb Z/_{2\mathbb Z} \longrightarrow 1$$ Using group representations $\mathbb Z/_{2\mathbb Z} \simeq \left\{\begin{pmatrix}-1&0\\0&-1\end{pmatrix}, \begin{pmatrix}1&0\\0&1\end{pmatrix}\right\}$ and $\mathbb Z/_{4\mathbb Z} \simeq \left\{\begin{pmatrix}-1&0\\0&-1\end{pmatrix},\begin{pmatrix}1&0\\0&1\end{pmatrix}, \begin{pmatrix}0&1\\-1&0\end{pmatrix}, \begin{pmatrix}0&-1\\1&0\end{pmatrix}\right\}$
There cannot be any retraction but using the surjection $\rho : \begin{pmatrix}a&b\\c&d\end{pmatrix} \longmapsto ad + bc$ isn't there a section $\iota : \begin{pmatrix}-1&0\\0&-1\end{pmatrix} \longmapsto \begin{pmatrix}0&-1\\1&0\end{pmatrix}$ ?
This would imply that we have a semi-direct product but I know this is not the case because $\mathbb Z/_{4\mathbb Z} \not\simeq \mathbb Z/_{2\mathbb Z} \rtimes \mathbb Z/_{2\mathbb Z} \simeq \mathbb Z/_{2\mathbb Z} \times \mathbb Z/_{2\mathbb Z}$... What gives ?
Your $\iota$ is not a homomorphism. $\iota(-I)^2 \ne \iota((-I)^2)$.