Set transporter problem

Given data
We are given a group $$G$$ acting on a set $$S$$ specified via a (small) generating set $$A$$ comprising permutations. We are given two subsets $$T_1$$ and $$T_2$$ of $$G$$.

Goal
We need to find whether there exists a $$g \in G$$ such that $$T_1^g = T_2$$.

Equivalent decision problems

 * Coset intersection problem: The reduction from set transporter to coset intersection follows directly from the fact that the set of elements in $$g$$ sending $$T_1$$ to $$T_2$$ is $$G \cap C$$ where $$C$$ is a coset of $$Stab_G(T_1)$$. The reduction the other way around involves looking at the action on $$S \times S$$ and asking whether there is an element sending the diagonal to a particular subset. (This idea is very similar to the idea used in showing that the group intersection problem and set-stabilizer problem are equivalent).


 * Group factorization problem: The equivalence of this follows easily from the fact that it is equivalent to the coset intersection problem.
 * Double coset membership testing problem: The equivalence of this again follows easily from the fact that it is equivalent to the coset intersection problem.
 * Set-stabilizer problem