Quadratic maps and Bockstein closed group extensions

Let $E$ be a central extension of the form $0 \to V \to G \to W \to 0$ where $V$ and $W$ are elementary abelian $2$-groups. Associated to $E$ there is a quadratic map $Q: W \to V$, given by the $2$-power map, which uniquely determines the extension. This quadratic map also determines the extension class $q$ of the extension in $H^2(W,V)$ and an ideal $I(q)$ in $H^2(G, \mathbb{Z} /2)$ which is generated by the components of $q$. We say that $E$ is Bockstein closed if $I(q)$ is an ideal closed under the Bockstein operator.

We find a direct condition on the quadratic map $Q$ that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map $Q_{\mathfrak{gl}_n}: \mathfrak{gl}_n (\mathbb{F}_2)\to \mathfrak{gl}_n (\mathbb{F}_2)$ given by $Q(\mathbb{A})= \mathbb{A} +\mathbb{A} ^2$ yield Bockstein closed extensions.

On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension $0 \to M \to \widetilde{G} \to W \to 0$ for some $\mathbb{Z} /4[W]$-lattice $M$. In this situation, one may write $\beta (q)=Lq$ for a ``binding matrix'' $L$ with entries in $H^1(W, \mathbb{Z}/2)$. We find a direct way to calculate the module structure of $M$ in terms of $L$. Using this, we study extensions where the lattice $M$ is diagonalizable/triangulable and find interesting equivalent conditions to these properties.