The authors construct new families of smooth admissible $overline{mathbb{F}}_p$-representations of $mathrm{GL}_2(F)$, where $F$ is a finite extension of $mathbb{Q}_p$. When $F$ is unramified, these representations have the $mathrm{GL}_2({mathcal O}_F)$-socle predicted by the recent generalizations of Serre's modularity conjecture. The authors' motivation is a hypothetical mod $p$ Langlands correspondence.