Equivariant homotopy equivalence of homotopy colimits of \(G\)-functors

Given a group G and a G-category \({\textbf{C}}\), we give a condition on a diagram of simplicial sets indexed by \({\textbf{C}}\) that allows us to define a natural action of G on its homotopy colimit, and some other simplicial sets defined in terms of the diagram. Well-known theorems on homeomorphisms and homotopy equivalences are generalized to equivariant versions.