  Given a finite group $G$ acting on a ring $R$, Merling constructed an
equivariant algebraic $K$-theory $G$-spectrum, and work of Malkiewich and
Merling, as well as work of Barwick, provides an interpretation of this
construction as a spectral Mackey functor. This construction is powerful, but
highly categorical; as a result the Mackey functors comprising the homotopy are
not obvious from the construction and have therefore not yet been calculated.
In this work, we provide a computation of the homotopy Mackey functors of
equivariant algebraic $K$-theory in terms of a purely algebraic construction.
In particular, we construct Mackey functors out of the $n$th algebraic
$K$-groups of group rings whose multiplication is twisted by the group action.
Restrictions and transfers for these functors admit a tractable algebraic
description in that they arise from restriction and extension of scalars along
module categories of twisted group rings. In the case where the group action is
trivial, our construction recovers work of Dress and Kuku from the 1980's which
constructs Mackey functors out of the algebraic $K$-theory of group rings. We
develop many families of examples of Mackey functors, both new and old,
including $K$-theory of endomorphism rings, the $K$-theory of fixed subrings of
Galois extensions, and (topological) Hochschild homology of twisted group
rings.
