Reconstructing under Group Actions

We give a bound on the reconstructibility of an action G[inline-graphic not available: see fulltext]X in terms of the reconstructibility of a the action N[inline-graphic not available: see fulltext]X, where N is a normal subgroup of G, and the reconstructibility of the quotient G/N. We also show that if the action G[inline-graphic not available: see fulltext]X is locally finite, in the sense that every point is either in an orbit by itself or has finite stabilizer, then the reconstructibility of G[inline-graphic not available: see fulltext]X is at most the reconstructibility of G. Finally, we give some applications to geometric reconstruction problems.