The flight instructor reported that during the helicopter instructional flight, hovering, straight-ahead, and 180-degree-turn autorotations were performed. The flight instructor's remedial actions typically included adjustment of pitch attitude to maintain the target airspeed, or collective adjustment to keep the rotor rpm in the normal operating range. During an autorotation with turn initiated by the student, the flight instructor took control of the helicopter but allowed the rotor rpm to decay below the lower limit. He adjusted collective and throttle, and initiated a decelerative flare to build rotor rpm, but the rotor rpm continued its decay. The flight instructor was unable to arrest the helicopter's descent, and the helicopter impacted the ground on the back of the skids, pitched forward and rolled over, which resulted in substantial damage to the fuselage, tailboom, and both the main and tail rotor systems. No preexisting mechanical anomalies were noted with the helicopter.