In this work, we present a high-power, high-repetition-rate, all-fiber femtosecond laser system operating at 1.5 $\unicode{x3bc}$ m. This all-fiber laser system can deliver femtosecond pulses at a fundamental repetition rate of 10.6 GHz with an average output power of 106.4 W – the highest average power reported so far from an all-fiber femtosecond laser at 1.5 $\unicode{x3bc}$ m, to the best of our knowledge. By utilizing the soliton-effect-based pulse compression effect with optimized pre-chirping dispersion, the amplified pulses are compressed to 239 fs in an all-fiber configuration. Empowered by such a high-power ultrafast fiber laser system, we further explore the nonlinear interaction among transverse modes LP₀₁, LP₁₁ and LP₂₁ that are expected to potentially exist in fiber laser systems using large-mode-area fibers. The intermodal modulational instability is theoretically investigated and subsequently identified in our experiments. Such a high-power all-fiber ultrafast laser without bulky free-space optics is anticipated to be a promising laser source for applications that specifically require compact and robust operation.