Pneumatic actuators have gained huge popularity in the field of soft robotics. A class of this kind of devices exploits inflatable thin membranes which generate a desired displacement upon inflation, but often without providing sufficient force/torque to perform their task. In this paper, we propose a novel actuator combining a membrane and a rigid foldable structure. Experimental tests show that such INFlatable ORigami Actuator (INFORA) is characterized by relatively high stiffness compared to other actuators of the same class. We provide a mathematical model to be used for design purposes and we describe the fabrication process. In addition, we show how the INFORA can be used to build a tendril-like structure capable of performing grasping tasks.