Preventing Usurpation of Power and Privileges by Extralegal Territories' Sedition Act of 2022 or the PUPPETS Act of 2022 This bill designates specified entities as foreign terrorist organizations and specially designated global terrorist organizations. The bill also prohibits the Department of Defense from taking any action that implies recognition of the sovereignty of such entities over territory or airspace claimed by the governments of Ukraine, Moldova, or Georgia. Entities subject to this bill include any entities holding themselves out to be the governments of any region of Ukraine that was not elected through elections in 2019 or 2020, the Pridnestrovian Moldovian Republic, the Republic of South Ossetia, or the Republic of Abkhazia.