# Constructive proof of Swan theorem

Let $$M$$ be an $$S_n$$-lattice (so it is free as an abelian group), and assume that $$M$$ is projective (i.e. direct summand of some $$\mathbb Z[S_n]^m$$). A theorem of Swan implies that $$M$$ is stably permutation, that is, $$M\oplus \mathbb Z[E]=\mathbb Z[F]$$, for some finite $$S_n$$-sets $$E$$ and $$F$$.

The theorems that go into the proof of this result are quite abstract and not constructive, I think. Is there a way to control $$E$$ and $$F$$? For example, can I always choose them so that no orbit has an $$A_n$$-stabilizer? Or such that every stabilizer is a $$p$$-group?