  We develop a version of $\mathrm{G}$-theory for an $\mathbb{F}_1$-algebra
(i.e., the $\mathrm{K}$-theory of pointed $G$-sets for a pointed monoid $G$)
and establish its first properties. We construct a Cartan assembly map to
compare the Chu--Morava $\mathrm{K}$-theory for finite pointed groups with our
$\mathrm{G}$-theory. We compute the $\mathrm{G}$-theory groups for finite
pointed groups in terms of stable homotopy of some classifying spaces. We
introduce certain Loday--Whitehead groups over $\mathbb{F}_1$ that admit
functorial maps into classical Whitehead groups under some reasonable
hypotheses. We initiate a conjectural formalism using combinatorial Grayson
operations to address the Equivariant Nishida Problem - it asks whether
$\mathbb{S}^G$ admits operations that endow $\oplus_n\pi_{2n}(\mathbb{S}^G)$
with a pre-$\lambda$-ring structure, where $G$ is a finite group and
$\mathbb{S}^G$ is the $G$-fixed point spectrum of the equivariant sphere
spectrum.
