Instructor pilot reported high RPMs after becoming airborne and an immediate descent to land on remaining runway length.

Narrative

Upon start up engine had some more vibrations than usual. Assuming this was due to the cold we proceeded to run up. Set power to approx 20% in the run up area to allow engine to warm up. Once oil temp reached 51 degrees pulled power back to idle. SP (Student Pilot) continued to run checklist and by the time we did full power check oil temp had reached 57 degrees. Oil temp; coolant; and gearbox oil were all in the green before full power check. First full power check we did not feel a prop switch. Load reached approx 94 and RPM's were 2200-2210. We double checked the allowed limits for RPM's in the afm and confirmed 2200-2300. Still due to no prop switch; we decided to do a second and third full power check. Both the second and third run up's had a prop switch but load and RPM stayed pretty close to the same; mid 90s load and low 2200's on RPM.Voter switch and ecu (Engine Control Unit) both checked good. Assuming the unusual run up and somewhat rough engine was due to the cold temps we continued to the runway for departure.Taxi was normal; students checklist usage was good and IP (Flight Instructor) verified everything was complete and ready for take off. We were cleared to take off runway XXL. The students introduction of power was smooth and normal. Load initially hit about 98% and RPMs were approx 2280. Soon after full power had been in a few seconds load had increased to 100 and RPMs to 2300. This immediately caught my eye because it was higher than what we were achieving over the course of 3 separate full power checks in the run up area. Load remained at 100 and RPM's continued to increase into the red (observed 2310). At this point the student was rotating and IP took controls and reduced power approx 3%. RPMs remained in the red and so I pulled power idle and landed the plane. I don't know exact engine readings after that as I was focused on flying the plane. Exact height reached also unknown but we were probably less than 20 feet in the air before we began our descent back to the runway.Once we were on the ground and full control of the airplane was established I called tower to notify them of the rejected take off. They said exit [taxiway] 1 which is where we exited. Joined [taxiway] 2; contacted ground and taxied back to lift.Plane was squawked.