  In [MMO] (arXiv:1704.03413), we reworked and generalized equivariant infinite
loop space theory, which shows how to construct $G$-spectra from $G$-spaces
with suitable structure. In this paper, we construct a new variant of the
equivariant Segal machine that starts from the category $\scr{F}$ of finite
sets rather than from the category ${\scr{F}}_G$ of finite $G$-sets and which
is equivalent to the machine studied by Shimakawa and in [MMO]. In contrast to
the machine in [MMO], the new machine gives a lax symmetric monoidal functor
from the symmetric monoidal category of $\scr{F}$-$G$-spaces to the symmetric
monoidal category of orthogonal $G$-spectra. We relate it multiplicatively to
suspension $G$-spectra and to Eilenberg-MacLane $G$-spectra via lax symmetric
monoidal functors from based $G$-spaces and from abelian groups to
$\scr{F}$-$G$-spaces. Even non-equivariantly, this gives an appealing new
variant of the Segal machine. This new variant makes the equivariant
generalization of the theory essentially formal, hence is likely to be
applicable in other contexts.
