On the ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$-equivariant intersection theory of $\mathrm{G}\mathrm{r}(2,4)$

We determine the ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$-equivariant Chow ring of $\mathrm{G}\mathrm{r}(2,4{)}^s$, the ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$-stable locus of $\mathrm{G}\mathrm{r}(2,4)$, over any algebraically closed based field of characteristic not equal to 2 or 3. In the process, we demonstrate that the quotient stack $[\mathrm{G}\mathrm{r}(2,4{)}^s\mathrm{/}{\mathrm{P}\mathrm{G}\mathrm{L}}_2]$ can be presented as the quotient of an open subset of ${\mathbb{P}}^1$ by a suitably chosen $S_4\le {\mathrm{P}\mathrm{G}\mathrm{L}}_2$. We also discuss some apparent difficulties with computing the full ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$-equivariant Chow ring of $\mathrm{G}\mathrm{r}(2,4)$.