The flight instructor reported that he was conducting a lesson on autorotation landings. After completing several autorotations with the student pilot, the flight instructor demonstrated three autorotation landings from a hover. On the student’s first attempt at the maneuver, the instructor reported that the student lowered the collective control instead of raising it. The flight instructor took control of the helicopter and applied up collective to help slow the descent rate. The helicopter began to yaw to the left. The helicopter landed on its left rear skid and bounced as it began to roll to the left. The instructor applied full right cyclic control to level the helicopter before touching back down on both skids. The bounced landing resulted in structural damage to the lower frame of the helicopter. The flight instructor reported no mechanical anomalies with the helicopter and associated flight control systems. The flight instructor stated that he anticipated that the student may apply down collective; however, his reaction time was not enough to counter the student’s control inputs.