  We study properties of continuous finite group actions on topological
manifolds that hold true, for any finite group action, after possibly passing
to a subgroup of index bounded above by a constant depending only on the
manifold. These include the Jordan property, the almost fixed point property,
as well as bounds on the discrete symmetry group. Most of our results apply to
manifolds satisfying some restriction such as having nonzero Euler
characteristic or having the integral homology of a sphere. For an arbitrary
topological manifold $X$ such that $H_*(X;{\mathbf Z})$ is finitely generated,
we prove the existence a constant $C$ with the property that for any continuous
action of a finite group $G$ on $X$ such that every $g\in G$ fixes at least on
point of $X$, there is a subgroup $H\leq G$ satisfying $[G:H]\leq C$ and a
point $x\in X$ which is fixed by all elements of $H$.
