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 $\mathcal{M}$ is a structure with universe $M$, and let $G$ be the automorphism group of $\mathcal{M}$. 
Define
$$V_0(M) = M, \quad V_{\alpha+1}(M) = V_\alpha\cup P(V_\alpha(M)), \quad V_\lambda = \bigcup_{\alpha<\lambda}V_\alpha(M)$$
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 $\mathcal{M}$, 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.