We introduce a family of impulse-like methods for the integration of highly oscillatory second-order differential equations whose forces can be split into a fast and a slow part. Methods of this family are specified by two weight functions: one is used to average positions and the other to mollify the force. In cases where the fast forces are conservative, the new family inlcudes as particular cases the mollified impulse methods introduced by García-Archilla, Skeel and the present author. On the other hand the methods here extend to nonlinear fast forces a well-known class of exponential integrators introduced by Hairer and Lubich. A convergence analysis will be presented that provides insight into the role played by the processes of averaging and mollification. A simple condition on the weight functions is shown to be both necessary and sufficient to avoid order reduction.