In this work, an all-fiberized and narrow-linewidth fiber amplifier with record output power and near-diffraction-limited beam quality is presented. Up to 6.12 kW fiber laser with the conversion efficiency of approximately 78.8% is achieved through the fiber amplifier based on a conventional step-index active fiber. At the maximum output power, the 3 dB spectral linewidth is approximately 0.86 nm and the beam quality factor is M_x² = 1.43, M_y² = 1.36. We have also measured and compared the output properties of the fiber amplifier employing different pumping schemes. Notably, the practical power limit of the fiber amplifier could be estimated through the maximum output powers of the fiber amplifier employing unidirectional pumping schemes. Overall, this work could provide a good reference for the optimal design and potential exploration of high-power narrow-linewidth fiber laser systems.