Reconstructing finite group actions and characters from subgroup information

A finite group G is said to be action reconstructible if, for any action of G on a finite set, the numbers of orbits under restriction to each subgroup always give enough information to reconstruct the action up to equivalence. G is character reconstructible if, given any matrix representation of G, the mean value of the character on each subgroup always gives enough information to reconstruct the character. The conjugacy matrix of G is the matrix whose (ij) entry is the number of elements of the jth conjugacy class belonging to a typical subgroup of the ith subgroup conjugacy class. It is shown that G is action reconstructible if and only if the rows of this matrix are linearly independent (which is in turn true if and only if G is cyclic), and is character reconstructible if and only if the columns are linearly independent (which is true if and only if any two elements of G which generate conjugate cyclic subgroups are themselves conjugate).