The relativistic klystron amplifier (RKA) is a very important kind of high power microwave device, which has the advantages of high power, high efficiency, stable output phase and amplitude. The development of multi-injection RKA toward engineering and practical application needs to further improve operating frequency and output power of klystron amplifier, while the RKA of conventional circular waveguide drift tubes is restricted by the physical factors such as geometric size, space charge force and high-voltage breakdown. The multi-beam RKA based on the technology of multiple electron beams can work at low voltage and guiding magnetic field, and can also possess high electron beam current and diversion coefficient. The physical limitation of conventional structure RKA is overcome, and the working frequency and the output power are improved. In the experiment, the X-band GW level power of microwave is obtained. The multi-beam RKA needs to further improve its working frequency and working life. In the experiment, the power conversion efficiency of multi-beam RKA is about 35%, and most of the remaining electron energy will accumulate on the collection pole at the end. If the heat dissipation of the collector is not designed appropriately, the collection will be seriously ablated when working at high heavy frequency. Thus a large quantity of plasma and secondary electrons are generated, which affects the stability of the device. To solve the problem of electron reflux bombarding the output cavity after electron beam exchanging energy in the gap of output cavity, the reflux process of relativistic electron beam in the device is analyzed in this paper. On this basis, a coaxial extraction structure with four-gap extension interaction is designed to avoid electron reflux and reduce the gap electric field, thus improving the working life of the device. At the same time, in order to solve the problem that the conventional water cooling channel can affect the output microwave mode in a high-frequency over-mode device, a mode transformation structure of coaxial TEM mode-fan-shaped TE₁₀ mode-coaxial TEM mode-circular waveguide TM₀₁ mode is designed. The mode conversion efficiency is greater than 99.9%, and the influence of collecting polar water cooling channel on the output microwave mode is avoided. The stable operation of multi-beam RKA in the X-band with a repetition rate of 45 Hz is realized experimentally, while the output power is over 1 GW and the microwave pulse width is over 100 ns. At present, the multi-beam RKA runs about 10000 times in total, and the output microwave parameters do not decrease significantly.