  The problem of equivariant rigidity is the $\Gamma$-homeomorphism
classification of $\Gamma$-actions on manifolds with compact quotient and with
contractible fixed sets for all finite subgroups of $\Gamma$. In other words,
this is the classification of cocompact $E_{fin}\Gamma$-manifolds.
  We use surgery theory, algebraic $K$-theory, and the Farrell--Jones
Conjecture to give this classification for a family of groups which satisfy the
property that the normalizers of nontrivial finite subgroups are themselves
finite. More generally, we study cocompact proper actions of these groups on
contractible manifolds and prove that the $E_{fin}$-condition is always
satisfied.
