The quickest way to get this quest is by killing Father then escaping through the central elevator. Find the Emergency Relay Button on the relay terminal then teleport out of The Institute to complete the quest.
When this quest starts while following a different faction's story quest, travel to The Institute, you'll only be inside the relay chamber while Father announces that you're an enemy of the Institute. After that, you'll be teleported outside of the CIT Ruins. The fast-travel map marker for the Institute will vanish from the map as well.