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There are several missions available when you play a CoD Ghosts’ campaign. However, you don’t need to complete all those missions to unlock Extinction mode. All you have to do is to complete the first four missions.  The missions that you need to complete are namely Ghost Stories, Brave New World, No Man’s Land, and Struck Down. You have to complete these missions on regular difficulty (or higher) to unlock Extinction mode. After completion, you just have to go back to the main menu and you will find that Extinction mode is already available.
Complete the first four missions. Go back to the Menu.