I would guess that the following is true, and that somebody has worked it out, but I don't recall ever seeing it. Can anyone point me to any literature on it?

Let $G$ be a finite group. We know that there is a functor 'equivariant suspension spectrum' $X\mapsto \Sigma^\infty_G(X_+)$ from $G$-spaces to $G$-spectra. It seems that there should be another such functor $A_G$, 'equivariant Waldhausen $K$-theory spectrum'. The fixed point spectrum ought to split up according to conjugacy classes of subgroups of $G$ just as for the suspension spectrum: $$ A_G(X)^G\sim \Pi_H\ A(X^H_{hW_GH}) $$ ($X^H_{W_GH}$ is homotopy orbits for the action of $W_GH=(N_GH)/H$ on fixed points of $H$.)

On the one hand when $X$ is based and connected then there should be a description of $A_G(X)$ that makes it a special case of a $K$-theory of structured $G$-ring-spectra.

On the other hand there should be a splitting $$ A_G(X)\sim \Sigma^\infty_G(X_+)\times Wh^{diff}_G(X) $$ where the other factor is related to equivariant pseudoisotopies and $h$-cobordisms.