Did you notice this article linked in the Roberts article?
https://aim4truth.org/2020/12/12/brett-kavanaugh-is-a-pilgrims-player/
I am reading the linked article. The Academy Society has no apparent connection to the Pilgrim’s society and I read the whole thing plus about their org.
I also read about the Inns of the Court and found no references to connections to the Pilgrim Society. I read all the links to the Inns of the Court orgs.
The claim that Kavanaugh is connected to the Pilgrims Society or that Barrett is a traitor are statements without any evidence at all.