Model theorists have a lot to say about so-called definable imaginary elements of a structure. One way to formulate imaginaries is the following: Suppose $\M$ is a structure with universe $M$, and let $G$ be the automorphism group of \M. Define V_0(M) = M, V_{\alpha+1}(M) = V_\alpha\cup P(V_\alpha(M)), V_\lambda = \bigcup_{\alpha<\lambda}V__\alpha(M) when \lambda is a limit ordinal. Let V(M) = \bigcup_{\alpha<\infty}V__\alpha(M). Then G has a natural action on $V(M)$ defined inductively by g(x) = {g(y):y\in x}. If x\in V(M) and S\subseteq M, then S supports x if G(S)\subseteq G({x}), where G(S) is the pointwise stabilizer of S and G({x}) is the setwise stabilizer of x. The imaginaries are those x\in V(M) which have finite support.

Now, for a finite structure \M, it makes sense to work with HF(M) instead of V(M) (stop the construction at the first countable infinite ordinal).

I would like to know if anyone has done any work regarding the action of a group of permutations of a finite set X on the hereditarily finite sets above X. Ideally, I'd like to get results "off the shelf" if they're out there.