Yes. In 2020, Natural England publised statutory guidance on trapping Stoats: "Stoats: licence to trap them to conserve wild birds (GL38)" and "Licence to trap stoats to prevent serious damage to livestock (GL39)" This guidance lists conditions under which GL38 and GL39 can be used, and incredibly it includes allowing gamekeepers to: "Use this licence [GL38] to conserve the following wild birds which are vulnerable to predation by stoats: game birds in situations when they’re vulnerable to stoat predation.."   And... "Use this licence [GL39] to prevent serious damage to the following livestock which are vulnerable to predation by stoats: domestic poultry and waterfowl game birds and wildfowl while held captive within a fenced pen"   So Natural England is indeed allowing the killing of Stoats to 'protect' birds reared solely to be shot for fun and profit.