I'll throw it out there as a rough concept for consideration:
Following a precautionary evacuation order by Father, a portion of The Institute's population was relocated to temporary quarters north of The Commonwealth in an abandoned Vault. They were provided with supplies, equipment, a contingent of Synths guards, and most critically, a backup of The Institute's archives for safekeeping.
In the wake of the destruction of The Institute by the Sole Survivor, and despairing at the loss of their home and many of their friends and families, these remnants made plans to seek safety far away from The Commonwealth. Somewhere along the line they became aware of the possible existence of Acadia, a Synth refugee that had managed to evade their gaze for quite some time. Seeing opportunity, and led by a new self-proclaimed Director, they traveled north to where it was believed to be located. Members of the former SRB and Robotics divisions soon made contact with DiMA under the pretense of being Synths which had managed to escape The Institute's destruction and were seeking a new home. DiMA quickly and gladly offered up Acadia's location to these supposed refugees.
Before arriving in force, the remnants sent ahead SRB agents and two Coursers. Upon their arrival, they mingled successfully with the populace of Far Harbor and soon came to the realization that Avery, the leader of the town, was in fact a Synth replacement: a replacement never authorized by the SRB.
Seeking to use this knowledge to their advantage, the new Director executed his plan. Utilizing what Synth troopers were still left, The Institute forces attacked Acadia and successfully overwhelmed it, capturing DiMA and subduing a substantial portion of the Acadian Gen-3's via their reset codes. With the former Synth sanctuary now under their control, they made overtures to the people of Far Harbor. While initially outraged by Acadia's destruction, Far Harbor came around quickly enough when The Institute offered its help not only against the Children of Atom, but to drive back the fog as well. Captain Avery also began to argue quite convincingly about how she'd never really trusted DiMA and how these 'new' neighbors were, at the very least, human. A suspicious change of heart to be sure.
In the few years that have passed, The Institute's remnants have managed to solidify their grip over Mount Desert Island and the town of Far Harbor. The Children of Atom have all but been driven out and the fog has been contained thanks to their reverse engineering and improvement upon DiMA's Fog Condensers. Slowly but surely, they are expanding their resources and founding a new life for themselves far away from The Commonwealth and the betrayal that brought them to the brink of destruction.
