The Chaos Insurgency has access to a more limited number of SCPs than the Foundation, but are willing to use them without restriction. The Insurgency also wants to take down the SCP Foundation for its 'lies', possibly by unleashing multiple Euclid or Keter-class SCPs and has provoked the Foundation on multiple occasions. The Chaos Insurgency, unlike the Foundation who easily 'store' their SCPs, would like to utilize these artifacts, most likely as weapons or torture devices.
The O5 Council, the council responsible for the creation of Insurgency, was in shock after hearing that their covert black ops team had defected, under their new name, 'Chaos Insurgency', as it's known today. The Foundation was alerted to this when Insurgency attacked multiple SCP facilities, stole multiple SCP objects, and evacuated Foundation defectors. Unbeknownst to the Foundation, Insurgency had been operating under the guise of Foundation defectors. The group was a covert special task force formed by the SCP Foundation, first known under the code-name of 'Insurgency', and was part of the Foundation.