If the patentee is a school, can the school not report it?

Of course. The patentee is the subject of the patent right, including the patentee and the obligee. The former can be citizens, collective ownership units, foreign trade enterprises and Sino-foreign joint ventures, while the latter is a unit owned by the whole people. If the patentee is a school, there is no need to report it to the school. If you follow the process of the school, you are the patent inventor, not the patent applicant. The patent applicant and the patentee enjoy the rights and interests of the patent.