Sheaves on Grothendieck constructions

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 $Sh_G(X)$.