  We show that, for a sheet or a Lusztig stratum S containing spherical
conjugacy classes in a connected reductive algebraic group G over an
algebraically closed field in good characteristic, the orbit space S/G is
isomorphic to the quotient of an affine subvariety of G modulo the action of a
finite abelian 2-group. The affine subvariety is a closed subset of a Bruhat
double coset and the abelian group is a finite subgroup of a maximal torus of
G. We show that sheets of spherical conjugacy classes in a simple group are
always smooth and we list which strata containing spherical classes are smooth.
