On the universal mod $p$ supersingular quotients for ${\mathrm{G}\mathrm{L}}_2(F)$ over ${\bar{\mathbb{F}}}_p$ for a general $F\mathrm{/}{\mathbb{Q}}_p$

Let $F\mathrm{/}{\mathbb{Q}}_p$ be a finite extension. We explore the universal supersingular mod $p$ representations of ${\mathrm{G}\mathrm{L}}_2(F)$ through computing a basis of their invariant space under the pro-$p$ Iwahori subgroup. This generalizes works of Breuil and Schein from ${\mathbb{Q}}_p$ and the totally ramified cases to the arbitrary one. Using these results we then construct for an unramified $F\mathrm{/}{\mathbb{Q}}_p$ a quotient of the universal supersingular module which has as quotients all the supersingular representations of ${\mathrm{G}\mathrm{L}}_2(F)$ with a ${\mathrm{G}\mathrm{L}}_2({\mathcal{O}}_F)$-socle that is expected to appear in the mod $p$ local Langlands correspondence. A construction for the case of an extension of ${\mathbb{Q}}_p$ with inertia degree 2 and suitable ramification index is also presented.