  Many three dimensional manifolds are two-fold branched covers of the three
dimensional sphere. However, there are some that are not. This paper includes
exposition about two-fold branched covers and many examples. It shows that
there are three dimensional homology spheres that do not two-fold branched
cover any manifold, ones that only two-fold branched cover the three
dimensional sphere, ones that just two-fold branched cover a non-trivial
manifold, and ones that two-fold branched cover the sphere and non-trivial
manifolds.
  When a manifold is surgery on a knot, the possible quotients via involutions
generically correspond to quotients of the knot. There can however be a finite
number of surgeries for which there are exceptional additional symmetries. The
included proof of this result follows the proof of Thurston's Dehn surgery
theorem. The paper also includes examples of such exceptional symmetries. Since
the quotients follow the behavior of knots a census of the behavior for knots
with less than eleven crossings is included.
