This paper deals with a class of nonlinear cascaded systems undergoing impulsive actions and an external input disturbance signal with bounded amplitude. Our main interests are to establish sufficient conditions to guarantee the input-to-state stability (ISS) and design a feedback \(H_{\infty }\) control. To do so, the system is decomposed into two cascaded (or interconnected) impulsive subsystems, namely the leader subsystem with the external input signal and follower subsystem with input being the output of the leader subsystem. We develop the ISS for each isolated impulsive subsystem and then for the cascaded system. We observed that the cascade of two ISS impulsive subsystems implies the ISS of the cascaded system. Later, these results are applied to establish another characterization of ISS where the controlled measured output of the leader is cascaded to the follower. Besides, the continuous leader subsystem is unstable but stabilized by an \(H_{\infty }\) controller. The Lyapunov method is used to achieve ISS where each subsystem admits an ISS-Lyapunov function, then the cascaded system is ISS with a Lyapunov function being the composite of the former functions. A numerical example and simulations are presented to enhance the findings of this work.