The headset is not compatible with the Xbox, even if you use the USB stick. The reason for this is the USB protocol used by the Xbox, which does not match the headset.

Yes, the USB stick (USB receiver) is included with the Logitech G535 LIGHTSPEED Wireless Gaming Headset. This receiver enables the wireless connection via LIGHTSPEED technology.

According to the information at our disposal, this model does not seem to offer a jack.