I couldn't show the underlined part. Can anyone explain it further?
$End_{\mathcal{O}}(M)$ is a G-algebra where $$g \cdot \psi = (x \mapsto g \psi(g^{-1}x))$$ and hence we have $$End_{\mathcal{O}}(M)^H = End_{\mathcal{OH}}(M)$$
Source (full book): G-algebras and modular representation theory (Jacques Thévenaz)