  We introduce a version of algebraic $K$-theory for coefficient systems of
rings which is valued in genuine $G$-spectra for a finite group $G$. We use
this construction to build a genuine $G$-spectrum
$K_G(\underline{\mathbb{Z}[\pi_1(X)]})$ associated to a $G$-space $X$, which
provides a home for equivariant versions of classical invariants like the Wall
finiteness obstruction. We provide a comparison between our $K$-theory spectrum
and the equivariant $A$-theory of Malkiewich--Merling via a genuine equivariant
linearization map.
