  Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let
$\mathrm{Sch}/\Bbbk$ denote the category of schemes of finite type over
$\Bbbk$. Let $B$ be a connected projective scheme over $\Bbbk$ and let
$\mathcal L$ be an ample line bundle on $B$. Let $\tau$ be a Harder-Narasimhan
type of length 2, and let $\delta\in\mathbb N$. We say a pure sheaf $\mathcal
E$ on $B$ is $(\tau,\delta)$-stable if its Harder-Narasimhan filtration
$0=\mathcal E_{\leq 0}\subsetneq\mathcal E_{\leq 1}\subsetneq\mathcal E_{\leq
2}=\mathcal E$ is non-splitting, of type $\tau$, with stable subquotients, and
$\delta=\dim_\Bbbk\mathrm{Hom}_{\mathcal O_B}(\mathcal E_2,\mathcal E_1)$ for
$\mathcal E_i:=\mathcal E_{\leq i}/\mathcal E_{\leq i-1}$.
  We define a moduli functor $\mathbf M'_{\tau,\delta}$ classifying
$(\tau,\delta)$-stable sheaves on $B$ and construct its coarse moduli space by
non-reductive geometric invariant theory (GIT). We extend the non-reductive GIT
in arXiv:1607.04181 and arXiv:1601.00340 to linear actions on non-reduced
schemes, and apply our non-reductive GIT to prove that the sheafification
$(\mathbf M'_{\tau,\delta})^\sharp$ on $(\mathrm{Sch}/\Bbbk)_{\'etale}$ is
represented by a quasi-projective scheme.
  Our methods generalise Jackson's construction of moduli spaces of
$(\tau,\delta)$-stable sheaves in arXiv:2111.07428 in the category of
varieties, to allow non-reduced moduli schemes.
