  In this paper we introduce a generalisation of a covariant Grothendieck
construction to the setting of sites. We study the basic properties of defined
site structures on Grothendieck constructions as well as we treat the
cohomological aspects of corresponding toposes of sheaves. Despite the fact
that the toposes of $G$-equivariant sheaves $Sh_G(X)$ have been introduced in
literature, their cohomological aspects have not been treated properly in a
desired fashion. So in the end of the paper we study some of the acyclic
families, introduce new type of acyclic resolutions which we call the
$G$-equivariant Godement resolutions, the degree of actions, and some other
basic cohomological concepts arising in $Sh_G(X)$.
