Integrated readout systems are desired to enable future large-scale superconducting quantum computers. These systems require high-performance cryogenic low-noise amplifiers, and implementing these in CMOS is desirable from an integration point of view. However, realizing the necessary noise and power performance required for this application while using CMOS is an open challenge. Here, we present the design of a cryogenic low-noise amplifier (LNA) in 22-nm fully depleted silicon on insulator (FDSOI) technology. Operating between 4 and 6 GHz and consuming 15.8 mW, it achieved a peak gain of 38 dB, a minimum noise of 3.5 K at 5.2 GHz, and an average noise of 4.4 K. Through back-gate control and bias optimization, it can be operated at a lower supply voltage while dissipating <4.5 mW at the expense of 0.7-K higher noise. Considering a figure of merit (FOM), which takes into account the number of added noise photons, gain, bandwidth, and power consumption, the LNA, biased at low power, demonstrates an FOM of >3× higher than other state-of-the-art cryogenic CMOS (cryo-CMOS) LNAs. To the best of our knowledge, this is the first report of a cryo-CMOS LNA operating above 4 GHz that exhibits a noise temperature below 4 K.