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In the stable category of bounded below $\mathcal{A}(1)$--modules, everymodule is determined by an extension between a module with trivial$Q_0$-Margolis homology and a module with trivial $Q_1$-Margolis homology. Weshow that all bounded below $\mathcal{A}(1)$-modules of finite type whose$Q_1$-Margolis homology is trivial are stably equivalent to direct sums ofsuspensions of a distinguished family of $\mathcal{A}(1)$-modules. Each modulein this family is comprised of copies of $\mathcal{A}(1) /\!/ \mathcal{A}(0)$linked by the action of $Sq^1 \in \mathcal{A}(1)$.  The classification theorem is then used to simplify computations of$h_0^{-1}\mathrm{Ext}_{\mathcal{A}(1)}^{\bullet, \bullet}\big(-,\mathbb{F}_2\big)$ and to provide necessary conditions for lifting$\mathcal{A}(1)$-modules to $\mathcal{A}$-modules. We discuss a Davis--Mahowaldspectral sequence converging to$h_0^{-1}\mathrm{Ext}_{\mathcal{A}(1)}^{\bullet, \bullet}(M, \mathbb{F}_2)$where $M$ is any bounded below $\mathcal{A}(1)$-module. The differentials inthis spectral sequence detect obstructions to lifting the$\mathcal{A}(1)$-module, $M$, to an $\mathcal{A}$-module. We give a formula forthe second differential.

Classifying and extending $Q_0$-local $\mathcal{A}(1)$-modules