After completing “Special Delivery”, you’ll have to find Emogene Cabot. It will be easier to ask around Goodneighbor’s Third Rail bar to get more information.
Head to the Third Rail and talk to Charlie first then ask Magnolia next. After getting your objective updated, head to the Charles View Ampitheater.
Talk to Brother Thomas and ask him about Emogene. He’ll claim that she won’t see anybody and you’ll have several options in your hands. You can offer 500 caps to see her or threaten Thomas. If your persuasion attempt failed, everyone will turn hostile. Kill them off and get the Ampitheater Key from Thomas’ body.
Head to the back of the stage and open the door using the Ampitheater Key. Talk to Emogene then return to the Cabot House to report the results.
When you arrive you’ll find Jack talking to Edward who is in the Asylum over the radio. After that, he’ll talk to you. Tell him that you found Emogene to complete the quest and start yet another one.