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Given a moduli problem posed using Geometric Invariant Theory, one can useNon-Reductive Geometric Invariant Theory to quotient unstable HKKN strata andconstruct 'moduli spaces of unstable objects', extending the usual moduliclassifications. After giving a self-contained account of how to do this, weapply this method to construct moduli spaces for certain unstable coherentsheaves of HN length 2 on a projective scheme, which we call $\tau$-stablesheaves. This extends a previous result of Brambila-Paz and Mata-Guti\'{e}rrezfor rank two vector bundles on a curve.

Moduli Spaces of Unstable Objects: Sheaves of Harder-Narasimhan Length 2