  For an atomic orbital base category in the sense of
Barwick-Dotto-Glasman-Nardin-Shah, we introduce the category of parametrised
perfect-stable categories and use it to construct the parametrised version of
noncommutative motives in which algebraic K-theory is corepresented.
Furthermore, we initiate a rudimentary theory of parametrised cubes which could
be of independent interest, generalising some of the elements in Dotto's theory
of equivariant Goodwillie calculus beyond the equivariant case. Using this
cubical theory, we show that in the equivariant case for finite 2-groups G, the
parametrised noncommutative motives canonically refine to G-symmetric monoidal
categories. Consequently, this endows the equivariant algebraic K-theory
spectra for these groups with the structure of E-infinity-ring spectra equipped
with multiplicative norms in the sense of Hill-Hopkins-Ravenel. Along the way,
we will also provide a machine to manufacture G-symmetric monoidal categories
from symmetric monoidal categories equipped with G-actions and elucidate how
the aforementioned parametrised perfect-stable categories relate to Mackey
functors valued in perfect-stable categories.
