  We prove that a certain genuine Hecke algebra $\mathcal{H}$ on the non-linear
double cover of a simple, simply-laced, simply-connected, Chevalley group $G$
over $\mathbb{Q}_{2}$ admits a Bernstein presentation. This presentation has
two consequences. First, the Bernstein component containing the genuine
unramified principal series is equivalent to $\mathcal{H}$-mod. Second,
$\mathcal{H}$ is isomorphic to the Iwahori-Hecke algebra of the linear group
$G/Z_{2}$, where $Z_{2}$ is the $2$-torsion of the center of $G$. This
isomorphism of Hecke algebras provides a correspondence between certain genuine
unramified principal series of the double cover of $G$ and the
Iwahori-unramified representations of the group $G/Z_{2}$.
