A mod-$p$ metaplectic Montréal functor

We extend Colmez’s functor defined for ${\mathrm{GL}}_2({\mathbf{Q}}_p)$ to the category of finitely generated smooth admissible mod-$p$ representations of the two-fold metaplectic cover of ${\mathrm{GL}}_2({\mathbf{Q}}_p)$. We compute the images of the absolutely irreducible genuine objects and obtain a bijection between the genuine supersingular representations and four-dimensional irreducible Galois representations invariant under twist by all characters of order two. Restricted to genuine objects, the extended functor naturally takes values in the category of what we call metaplectic Galois representations – Galois representations with a certain extra structure encoding the aforementioned twist-invariance.