Finite groups of OD-conjugates

Abstract

Given any set x, define the set of conjugates of x to be Given any subgroup G of the group of permutations of {1, ⋯, n}, it is consistent with ZFC that there exists an ordered n-tuple 〈 x ₁, ⋯, x_n〉 such that

$$c(\langle x_1 ,...,x_n \rangle ) = \{ \langle x_{\pi 1} ,...,x_{\pi n} \rangle |\pi \in G\} .$$