Given a group action from G to G/H, show that N(H)/H is isomorphic to G/H

1. Nov 13, 2012

potmobius

1. The problem statement, all variables and given/known data

G is a group, H is a subgroup of G, G acts on G/H in the standard manner, and N(H) is the normalizer of H in G. Show that there is an isomorphism between Aut(G/H) and N(H)/H, where Aut(G/H) is the set of G-equivariant bijections f:G/H -> G/H

2. Relevant equations

3. The attempt at a solution

I know from a previous theorem that there exists an isomorphism from Aut(X) to X, so here its only necessary to prove that G/H and N(H)/H are isomorphic. Since G is the entire group in this case, it is also a normal group that contains H. So it can potentially be N(H). But I don't how to to guarantee an isomorphism.