  In this paper, we show that for any reductive group $G$ the moduli space of
semistable $G$-Higgs bundles on a curve in characteristic $p$ is a twisted form
of the moduli space of semistable flat $G$-connections. This is the semistable
version of a previous result of Chen-Zhu, and the $G$-bundle version of a
previous result of de Cataldo-Groechenig-Zhang. As a consequence, we show that
the Decomposition Theorem for the Hitchin morphism for $G$-Higgs bundles has
the same shape as that for the de Rham-Hitchin morphism for flat
$G$-connections.
