This quest to investigate a member of the Children is given by the High Confessor after you speak to him and is part of the What Atom Requires quest. Read his note to see who you will be investigating.
See if there's any truth behind the High Confessor's fears of Sister Auberts treachery. Head downstairs into the bulkhead of the submarine to find her bed. There will be a note hidden under the corner of her mattress.
Follow the waypoint marker to the room on the left when you first enter Nucleus to find the key to the footlocker Edgar and Sister Aubert used for their correspondences.
Follow the next waypoint marker back down into the submarine to discover the access the footlocker and an implicating note from Sister Aubert to Edgar.
You can optionally speak to Sister Aubert before reporting back to the High Confessor. Report back to the High Confessor. Pass a speech check to convince him that Aubert's loyal and receive the Inquisitor's Cowl, a helmet that increases the wearer's intelligence with increasing Rads, as a reward. Note that if you instead choose the option to tell the High Confessor that you didn't find anything, you will not receive the cowl.