They were sent in as the 2nd squad by the U.S. Military to pacify the "alien" outbreak in Caldera Peak, Colorado after CIF Team Four of the Rapid Reaction Force had mysteriously disappeared within the first hour of deployment. The Task Force's main task was to drill and destroy all of the Cryptid hives and ultimately, fight their way through the heart of the post-apocalyptic, Cryptid-infested town, in order to detonate the nuke (authorzied by the President of the United States) to completely eradicate the place of Cryptids. The town had become highly infested with Cryptids due to the ODIN strikes that had occured 2 weeks previously, which revealed the the prehistoric Cryptid species living in a colony underneath the nearby Colorado colonies. After they had activated the countdown, the Task Force quickly rushed back to the extraction point, while fighting off an onslaught of Cryptids, where they exfil by the helicopter, and made it just in time as they watched from the helicopter, the nuke sterilizing the town as well as all Cryptid presence in the area.
The status of Task Force Spectre after they nuked the town is currently unknown.
The Task Force's callsign may be a reference to Specter Team, who served as the protagonist in the Co-op campaign in Resistance 2, as Extinction bears many resemblances and possible inspirations from Resistance 2's Co-op campaign.