They are the stabilizers in G.

Actually, I decided to ignore the advice about the 2nd isomorphism theorem and I thought of the following map that I think is a bijection.

the orbit containing g(a), where a is given in the problem.

The map is definitely surjective. It is well defined because if you have two cosets

and

then it must be true that

for elements

and

where the 2nd equality is because g_a is a stabilizer and the last equality because in part (a), it is shown that any element of G permutes the different orbits, taking all elements in one orbit to all elements in another...

Finally it is injective because if

then

This seems to work... perhaps the hint was a red-herring (or perhaps I made a mistake).