The following came up in a problem on graph reconstruction. It isn't very important, but I thought some people here might find it interesting and not too trivial (I'm not a group theorist).

Take a set $\Omega_n=\lbrace x_1,\ldots,x_n,y_1,\ldots,y_n\rbrace$ of $2n$ distinct atoms. A *partition of $\Omega_n$ into ordered pairs* is a set of ordered pairs of the form $x_iy_i$ or $y_ix_i$ such that for each $i$, exactly one of $x_iy_i$ and $y_ix_i$ appears.
There are exactly $2^n$ such partitions altogether. An example of a partition of $\Omega_3$ into ordered pairs is $\lbrace x_1y_1,y_2x_2,x_3y_3\rbrace$,
which is different from $\lbrace x_1y_1,y_2x_2,y_3x_3\rbrace$.

Now let $G$ be a permutation group on $\Omega_n$. $G$ has a natural action on the set of all partitions of $\Omega_n$ into ordered pairs: just apply it to the atoms where they appear. Thus, for $g\in G$, $\lbrace x_1y_1,y_2x_2,x_3y_3\rbrace^g = \lbrace x_1^gy_1^g,y_2^gx_2^g,x_3^gy_3^g\rbrace$. Now suppose that the set of partitions is closed under this action and that this action is *transitive* on the partitions.

**Question:** What can we say about $G$?

(Added:) Since we want $G$ the set of partitions to be closed under $G$, we know that $G$ is a subgroup of the wreath product $Z_2\wr S_n$.

Obviously $|G|\ge 2^n$. Also, it suffices to consider (inclusion-)minimal groups with the required transitivity.

The easiest example, with order, $2^n$ is $$G=\langle (x_1\,y_1), (x_2\,y_2), \ldots, (x_n\,y_n)\rangle.$$ Slightly less obviously, but still with order $2^n$, divide $\lbrace 1,2,\ldots,n\rbrace$ into singletons and pairs, and have a transposition for the singletons and a 4-cycle for the pairs. This is an informal definition, but as an example for $n=6$: $$G=\langle (x_1\,x_2\,y_1\,y_2), (x_3\,y_3), (x_4\,y_4), (x_5\,x_6\,y_5\,y_6)\rangle.$$ What else is there?