Set-stabilizer in a 2-group problem

Given data
A 2-group $$G$$ acts faithfully on a set $$S$$, and it is described by a generating set $$A$$. We are given a subset $$T \subseteq S$$.

Goal
We need to find the set-stabilizer of $$T$$ in $$G$$, that is, the set of elements of $$G$$ that stabilize $$T$$.

Outline

 * 1) Construct a structure forest for $$G$$. Such a structure forest must be a union of complete binary trees.
 * 2) Now, pair together structure trees of the same size, to get a complete binary tree with depth 1 more. Keep doing this pairing procedure (like binary addition) till all the trees have different sizes
 * 3) Thus, obtain trees of different sizes. The automorphism group of this new forest is a Sylow 2-subgroup of $$Sym(n)$$. Call this Sylow subgroup $$P$$.
 * 4) For all the vertices in $$T$$, attach a long gadget (say a path) from these vertices, in such a way that the only permissible automorphisms of the new tree (after attaching these gadgets) are automorphisms that send vertices of $$T$$ to among themselves.
 * 5) Having put these gadgets, compute the automorphism group of the forest along with these gadgets. This can easily be done as it is a special case of finding the automorphism group of a rooted forest. Call this automorphism group $$M$$. (the automorphism group of a rooted forest can be found by the rooted tree isomorphism problem.

We now have a situation where:


 * $$G$$ sits inside a 2-Sylow subgroup $$P$$
 * There exists a subgroup $$M$$ containing the whole of $$Stab_G(T)$$ that also sits inside $$P$$ and such that $$G \cap M = Stab_G(T)$$
 * Sicne $$P$$ is a 2-group, both $$G$$ and $$M$$ are subnormal in $$P$$. Thus, computing $$G \cap M$$ reduces to the subnormalizing group intersection problem.