The PS5 controller needs to be wired and the Bluetooth must be turned off in order to play the PS5 version of the game on Call of Duty.
Bluetooth Controller Adapter for PS4/ PS5/ Switch Pro Controller, AOLION Wireless Controller Adapter Compatible with Switch/ Switch Lite/ Windows PC/ Android TV Box/ Steam (Include a Type-c OTG Cable) : Video
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Cronus Zen Controller Emulator for Xbox, Playstation, Nintendo and PC (CM00053) : Video Games
Beloader PS5 Keyboard and Mouse Converter Guide.PS5 Beloader Support Cronus Zen,Xim Apex,Resnow S1,Titan Two with Playstation 4 KBM and Controller to Play All PS5 FPS Games