  Let $p$ be an odd prime. Let $F$ be a non-archimedean local field of residue
characteristic $p$, and let $\mathbb{F}_q$ be its residue field. Let
$\mathcal{H}^{(1)}_{\mathbb{F}_q}$ be the pro-$p$-Iwahori-Hecke algebra of the
$p$-adic group ${\textrm GL_2}(F)$ with coefficients in $\mathbb{F}_q$, and let
$Z(\mathcal{H}^{(1)}_{\mathbb{F}_q})$ be its center. We define a scheme
$X(q)_{\mathbb{F}_q}$ whose geometric points parametrize the semisimple
two-dimensional Galois representations of ${\textrm Gal}(\overline{F}/F)$ over
$\overline{\mathbb{F}}_q$. Then we construct a morphism from the spectrum of
$Z(\mathcal{H}^{(1)}_{\mathbb{F}_q})$ to $X(q)_{\mathbb{F}_q}$ generalizing the
morphism appearing in \cite{PS2} for $F=\mathbb{Q}_p$. In the case
$F/\mathbb{Q}_p$, we show that the induced map from Hecke modules to Galois
representations, when restricted to supersingular modules, coincides with
Grosse-Kl\"onne's bijection \cite{GK18}. For this, we determine the Lubin-Tate
$(\varphi,\Gamma)$-modules associated to absolutely irreducible Galois
representations.
