  Let $G$ be a finite group acting on a small category $I$. We study functors
$X \colon I \to \mathscr{C}$ equipped with families of compatible natural
transformations that give a kind of generalized $G$-action on $X$. Such objects
are called $G$-diagrams. When $\mathscr{C}$ is a sufficiently nice model
category we define a model structure on the category of $G$-diagrams in
$\mathscr{C}$. There are natural $G$-actions on Bousfield-Kan style homotopy
limits and colimits of $G$-diagrams. We prove that weak equivalences between
point-wise (co)fibrant $G$-diagrams induce weak $G$-equivalences on homotopy
(co)limits. A case of particular interest is when the indexing category is a
cube. We use homotopy limits and colimits over such diagrams to produce loop
and suspension spaces with respect to permutation representations of $G$. We go
on to develop a theory of enriched equivariant homotopy functors and give an
equivariant "linearity" condition in terms of cubical $G$-diagrams. In the case
of $G$-topological spaces we prove that this condition is equivalent to
Blumberg's notion of $G$-linearity. In particular we show that the
Wirthm\"{u}ller isomorphism theorem is a direct consequence of the equivariant
linearity of the identity functor on $G$-spectra.
