I told him exactly that. He said:
Consider the case of chlorine.When we use FeCl3, it pulls Cl- weaker than AlCl3 and so, as a result, Cl is partially attached to the alkyl group still clinging with it, ever after it was taken by FeCl3. The carbocation is not fully formed and as a result, this does not allow rearrangement to take place.
I found this rather surprising and decided to ask the forum experts.