Frequency conversion in glass optical fibers requires both phase and group velocity matching between the pump and the higher harmonic when working with short pulses. We show that phase and group velocities can be matched simultaneously for third order nonlinear processes, by considering that the third harmonic propagates in the higher order azimuthally symmetric LP₀₃-mode. Moreover, the pump and frequency tripled signals can form an intermodal two-color pump to trigger a cascaded wave mixing process, which generates the second harmonic LP₀₁-mode. This opens avenues for second harmonic generation without need for a second order nonlinearity in the optical fiber.