Does the home office have any connection with the inland revenue?
Not directly, but they can access each other’s information if necessary – in theory anyway. The Home office runs the police, courts, prisons, etc, and also immigration, so they deal with student visas. Inland Revenue is taxation of all kinds, including Customs and Excise. Foreign students who are allowed to work and who earn enough to reach the level where they pay tax will have dealings with them.