  Let $\mathbf{G}$ be a connected reductive group over a {non-archimedean local
field} $F$. Let $K_\mathcal{F}$ be the parahoric subgroup attached to a facet
$\mathcal{F}$ in the Bruhat--Tits building of $\mathbf{G}$. The ultimate goal
of the present paper is to describe the center of the parahoric--Hecke algebra
$\mathcal{H}(\mathbf{G}(F)//K_{\mathcal{F}}, \mathbb{Z}[q^{-1}])$ with level
$K_{\mathcal{F}}$ and prove the compatibility of generalized (twisted)
Bernstein and Satake homomorphisms.
