A GIT construction of moduli spaces of sheaves of length 2

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.