  We extend Colmez's functor defined for $\operatorname{GL}_2(\mathbf{Q}_p)$ to
the category of finitely generated smooth admissible mod-$p$ representations of
the two-fold metaplectic cover of $\operatorname{GL}_2(\mathbf{Q}_p)$. 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.
