We know that permutations, elements of the symmetric group on a finite set with n elements, are conjugate iff they have the same cycle structure.
My question is that given two permutations that are conjugate in the symmetric group, can we find all the element that gives the conjugation? Also, can we find an element for conjugation which also fixes some another permutation, that is, two permutations having the same cycle structure, can we know whether they are conjugate via an element from the stabilizer of another permutation?
A trivial example, when we have 3 objects, (12) and (13) cycles are conjugate, yet there is no element in the stabilizer of (12) which does conjugation.
Another example is that when n = 4 , (12) and (13) are conjugate, yet there is no element in the stabilizer of (1234) with which when we conjugate (13), we obtain (12).
Thank you in advance, any kind of help is appreciated.