The stabilizing subgroup H(x) is the subgroup of G whose actions leave x fixed.
If a takes x to y and 1/a takes y to x, and H stabilizes x, then a*H/a stabilizes y. (Remember the backwards convention, apply a inverse, then H, then a.) Reverse this map to show stabilizing members for x and y corespond 1-1. The stabilizers for any two elements in the same orbit are isomorphic, via an inner automorphism on G. Conversely, the a-conjugate of the stabilizing subgroup of x stabilizes another element in the orbit of x, namely a(x).
For finite groups, |H|×|x| = |G|, where |x| is the size of the orbit of x.
If the stabilizer H is a normal subgroup of G, it is not changed by conjugation, hence it stabilizes everything in the orbit of x.