There is a subject of equivariant stable homotopy theory. You are referring to $bu$ as what is called a naive $G$-spectrum (a spectrum acted on by a group), and your notation $E^G$ is confusing the categorical fixed point spectrum with the homotopical fixed point spectrum. The difference between the two is, for example, the subject of the Segal conjecture.  There is a homotopy fixed point spectral sequence for homotopical fixed point spectra. There is a huge literature on equivariant stable homotopy theory (which was a key input to solution of the Kervaire invariant problem by Hill, Hopkinds, and Ravenel). One expository introduction is: Greenlees and May. Equivariant stable homotopy theory. Handbook of Algebraic Topology, edited by I.M. James, pp. 279-325. 1995.  But a lot has been done since then.  Naive $G$-spectra are not so interesting.  Genuine $G$-spectra have $RO(G)$-graded homology and cohomology.