  An isovariant map is an equivariant map between $G$-spaces which strictly
preserves isotropy groups. We consider an isovariant analogue of
Klein--Williams equivariant intersection theory for a finite group $G$. We
prove that under certain reasonable dimension and codimension conditions on
$H$-fixed subspaces (for $H\leq G$), the fixed points of a self-map of a
compact smooth $G$-manifold can be removed isovariantly if and only if the
equivariant Reidemeister trace of the map vanishes. In doing so, we study
isovariant maps between manifolds up to isovariant homotopy, yielding an
isovariant Whitehead's theorem. In addition, we speculate on the role of
isovariant homotopy theory in distinguishing manifolds up to homeomorphism.
