I have been thinking about the woman that Kate gives Aaron to. What if she was not really Claire's mother and this was a way for Ben or Eloise or TPTB to nab Aaron and get him back to the Island. Maybe Aaron is on the Island and we do not know it yet. Claire's mom was in very bad condition last we seen her and nobody ever questioned her on her identity when she was in LA. There was also nobody there that would be able to recognize her. For all we know Claire's mother may be deceased.