Induction can indeed be undertaken in the private sector but not all private schools offer it. After using up the 16 month supply allowance, you can apply for an LA extension and/or seek work int he private sector (any age group). There is no supply limit or other restriction on tecahing in the private sector as it is not even obligatory to have QTS to be a teacher there! A well regarded Head of a top private school took early retirement a few years ago and tried to offer his services as a teacher in the State sector. He was unsuccessful as he didn't have QTS.