  A central extension of the form $E: 0 \to V \to G \to W \to 0$, where $V$ and
$W$ are elementary abelian 2-groups, is called Bockstein closed if the
components $q_i \in H^*(W, \FF_2)$ of the extension class of $E$ generate an
ideal which is closed under the Bockstein operator. In this paper, we study the
cohomology ring of $G$ when $E$ is a Bockstein closed 2-power exact extension.
The mod-2 cohomology ring of $G$ has a simple form and it is easy to calculate.
The main result of the paper is the calculation of the Bocksteins of the
generators of the mod-2 cohomology ring using an Eilenberg-Moore spectral
sequence. We also find an interpretation of the second page of the Bockstein
spectral sequence in terms of a new cohomology theory that we define for
Bockstein closed quadratic maps $Q : W \to V$ associated to the extensions $E$
of the above form.
