While practicing hovering autorotations on an asphalt taxiway, several hard landings occurred as a result of the student's difficulty learning the proper application of collective pitch. The CFI then instructed the student to move to a sod area adjacent to the taxiway for the cushioning effect to reduce impact forces on the aircraft. The CFI said he did not inspect the aircraft for damage after the hard landings on the taxiway. The student then performed one hovering autorotation on the sod area, and during the subsequent hover control was lost. The aircraft rolled over and impacted the ground. Heavy skid impact marks were found on the taxiway. The engine mount tubular structure at the bottom mount, and the cross member support tubes, were fractured. The walls of the tubes were of irregular thickness, contained extensive corrosion scaling, and in some places could be bent by hand. The bottom of the engine, which is mounted to the transmission and main rotor mast, was free to move. Last annual insp. five days before accident.