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These spaces are related to $2$-Sylow subgroups of $S_n$. For example, if $n=2^k$, then $X^n_k$ is the set of $2$-Sylow subgroups of $S_n$. To see this, note that $S_n$ acts transitively on $X^n_k$, and that the stabilizer of any fixed element of $X^n_k$ is a $2$-Sylow subgroup of $S_n$. I don't really have a reference for this, but it is not too hard to work out the order of the automorphism group, and see that it is $2^{v_2(n!)}$ using the fact that $$ v_2(n!) = \lfloor n/2 \rfloor + \lfloor n/4 \rfloor + \dotsb $$