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