Theory is developed to predict the flows induced by small-amplitude two-dimensional vertically propagating internal wave packets under the influence of rotation. While the long-wave response originally predicted by Bretherton [J. Fluid Mech. 36, 785 (1969)JFLSA70022-112010.1017/S0022112069001984] dominates if the influence of background rotation is negligible, the induced waves are found to be evanescent if the Coriolis parameter is sufficiently large. Explicitly, evanescent induced flows dominate if the Coriolis parameter is greater than the forcing frequency, which is set by the ratio of the vertical group velocity of the wave packet divided by the vertical scale of modulation of the wave packet. The predicted structure of the induced flow is confirmed by numerical simulations. For an upward and rightward propagating Gaussian wave packet that dominantly induces a long-wave response, the flow is rightward across the upper flank and leftward across the lower flank of the wave packet. In contrast, the induced flow for a dominantly evanescent response is negative over the middle of the wave packet. This qualitative change in structure is anticipated to influence the modulational stability of moderately large-amplitude wave packets.