  Let $G$ denote a possibly discrete topological group admitting an open
subgroup $I$ which is pro-$p$. If $H$ denotes the corresponding Hecke algebra
over a field $k$ of characteristic $p$ then we study the adjunction between
$H$-modules and $k$-linear smooth $G$-representations in terms of various model
structures. If $H$ is a Gorenstein ring we single out a full subcategory of
smooth $G$-representations which is equivalent to the category of all
Gorenstein projective $H$-modules via the functor of $I$-invariants. This
applies to groups of rational points of split connected reductive groups over
finite and over non-archimedean local fields, thus generalizing a theorem of
Cabanes. Moreover, we show that the Gorenstein projective model structure on
the category of $H$-modules admits a right transfer. On the homotopy level the
right derived functor of $I$-invariants then admits a right inverse and becomes
an equivalence when restricted to a suitable subcategory.
