Reconstructing under Group Actions

TL;DR: A bound on the reconstructionibility of an action GX in terms of the reconstructibility of a the action NX, where N is a normal subgroup of G, and the reconstructability of the quotient G/N is given.

Abstract: We give a bound on the reconstructibility of an action G** X in terms of the reconstructibility of a the action N** 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** 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** X is at most the reconstructibility of G. Finally, we give some applications to geometric reconstruction problems.