  Let $F/\mathbb{Q}_p$ be a finite extension. We explore the universal
supersingular mod $p$ representations of $\mathrm{GL}_2(F)$ 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 $F/\mathbb{Q}_p$ a quotient of the universal supersingular module
which has as quotients all the supersingular representations of
$\mathrm{GL}_2(F)$ with a $\mathrm{GL}_2(\mathcal{O}_F)$-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.
