Reconstructing under Group Actions

We give a bound on the reconstructibility of an action GX in terms of the reconstructibility of a the action NX, where N is a normal subgroup of G, and the reconstructibility of the quotient G/N. We also show that if the action GX is locally finite, in the sense that every point is either in an orbit by itself or has finite stabilizer, then the reconstructibility of GX is at most the reconstructibility of G. Finally, we give some applications to geometric reconstruction problems.