In the framework of the Compact Light XLS project, a short ultra-high gradient linearizer working on the third harmonic of the main LINAC frequency is requested. Increasing gradients and reducing dimensions are requirements for XLS and all next generation linear accelerators. Actually, ultra-compact normal conducting accelerating structures, operating in the Ka-band regime ranging from 100 to 150 MV/m are required to achieve ultra-high gradients for research, industrial and medical applications. To fulfill these strong requirements, the R$\&$D of a proper Ka-band klystron with RF power output and a high efficiency is mandatory. This contribution reports the design of a possible klystron amplifier tube that provides and output signal at 36 GHz to feed the phase space linearizer, while receiving at the input, a low level signal oscillating at 12 GHz; which is the LINAC frequency. The proposed structure reaches an efficiency of 42$\%$ and a 20 MW RF power output. This contribution discusses also the high-power DC gun, the beam focusing channel and the RF beam dynamics.