In this work we derive a generalized map which describes the time evolution of a quantum system coupled to a quantum environment composed by a finite number of free particles kicked harmonically at periodic times. Dissipation is introduced via the interaction between system and environment, which is switched on and off simultaneously at regular time intervals. The dynamics of the environmental particles occurs along rotated ellipsis in phase space and can be described by unrotated harmonic oscillators with a new frequency which depends on the kicking time. With the definition of a thermal bath we show that our quantum kicked environment induces an unusual fluctuation–dissipation relation.