Spoiler
While recovering the amulet, you have to infiltrate the Mythic Dawn base at Lake Arrius Caverns peacefully, give up your possessions, listen to Mankar Camoran's speech, execute a helpless prisoner when prompted by Ruma Camoran, and you become an acolyte of the Mythic Dawn according to the factions page of your journal
However, doing so doesn't seem to get you much. SPOILER:
Spoiler
you're locked in behind a gate that requires a key and there's no key on any of the NPCs or in any of the containers you can reach. The gate will only open when you take the Mysterium Xarxes, which any NPC will tell you may only be touched by Mankar Camoran. As soon as you take it (even if the sneak meter indicates no one can see you), you're kicked out of the faction for your blasphemy and anyone in the room will attack you. I managed to sneak a few useful items into the shrine with me (by pickpocketing them back from Harrow), but you still get caught even if you are invisible or have 100% chameleon. Same thing happens if you kill any of the potential witnesses in the shrine (again, even if you're invisible)
Of course, I could teleport away with a console code, but does anybody have any other ideas on how to leave and avoid getting kicked out of the Mythic Dawn without cheating?