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The development of sensors working in a large range of temperature is of crucial importance in areas such as monitoring of industrial processes or personal tracking using smart objects. Devices integrating GaN/Ga₂O₃ core/shell nanowires (NWs) are a promising solution for monitoring carbon monoxide (CO). Because the performances of sensors primarily depend on the material properties composing the active layer of the device, it is essential to control them and achieve material synthesis in the first time. In this work, we investigate the synthesis of GaN/Ga₂O₃ core-shell NWs with a special focus on the formation of the shell. The GaN NWs grown by plasma-assisted molecular beam epitaxy, are post-treated following thermal oxidation to form a Ga₂O₃-shell surrounding the GaN-core. We establish that the shell thickness can be modulated from 1 to 14 nm by changing the oxidation conditions and follows classical oxidation process: A first rapid oxide-shell growth, followed by a reduced but continuous oxide growth. We also discuss the impact of the atmosphere on the oxidation growth rate. By combining XRD-STEM and EDX analyses, we demonstrate that the oxide-shell is crystalline, presents the β-Ga₂O₃ phase, and is synthesized in an epitaxial relationship with the GaN-core.