Trump fully had the authority to remove Mueller. That he didn’t was both good and bad. Good because he knew he had nothing to hide and it would have been taken as an admission of guilt if he had, and bad because he allowed Mueller or whoever he was a figurehead for to draw a six month investigation into 3 years.
Everything related to the oversight of a special counsel is internal to the U.S. Justice Department, which means it comes entirely under the authority of the President.