  Suppose that $\tilde{G}$ is a connected reductive group defined over a field
$k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\tilde{G}$
satisfying a certain quasi-semisimplicity condition. Then the connected part of
the group of $\Gamma$-fixed points in $\tilde{G}$ is reductive. We axiomatize
the main features of the relationship between this fixed-point group and the
pair $(\tilde{G},\Gamma)$, and consider any group $G$, not just the
$\Gamma$-fixed points of $\tilde{G}$, satisfying the axioms. (In fact, the
axioms do not require $\Gamma$ to act on all of $\tilde{G}$.) If both
$\tilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals
$\tilde{G}^*$ and $G^*$. We show the existence of and give an explicit formula
for a natural map from semisimple stable conjugacy classes in $G^*(k)$ to those
in $\tilde{G}^*(k)$. If $k$ is finite, then our groups are automatically
quasisplit, and our result specializes to give a map from semisimple conjugacy
classes in $G^*(k)$ to those in $\tilde{G}^*(k)$. Since such classes
parametrize packets of irreducible representations of $G(k)$ and
$\tilde{G}(k)$, one obtains a mapping of such packets.
