When you awake from being knocked out, you will be locked up and interrogated. Refuse to deal with Atumn; eventually the President of the Enclave will interrupt and allow you to leave and meet with him. On your way to the meeting, you will hear a command over the loudspeaker that you are to be killed. Prepare for some heavy fighting on your way to the President. Follow the mission marker on your compass until you finally get to the President.
During this conversation, you can attempt to convince the President to self-destruct. It doesn't really matter either way. After the conversation, grab the FEV virus he offers and leave the base. Head back to the Brotherhood Citadel to speak with Elder Lyons. You can give up the FEV during this coversation or hold on to it; again, doesn't matter much either way.
Note: Make sure to grab the bobblehead in Colonel Autumn's office before speaking with the President. You won't be able to get it after this.