This quest begins in Gridania at the Adders' Nest. Speak with Fulke to begin. He will discuss information regarding the Grand Company, and give you a task to deal with a Thaumaturge in the forest.
Travel to Quarrymill in the South Shroud (45,47) to confront Ailith. While Ailith will not leave yet, she provides you a task that may rectify the issue the Adders' Nest has with her. She requests you deal with another Thaumaturge who has been traveling in the forest, Urianger.
After speaking with Ailith, travel southwest from Quarrymill to the 2nd level of South Shroud (42,48). Here, you will confront Urianger, the Thaumaturge Ailith spoke of. Upon speaking with him, he will ask if you would take up his cause. Selecting "No" will decline, and he will spawn a Drake for you to dispatch. You have 30 minutes to complete this instanced battle with the Drake.
Upon dispatching the Drake, you will be met by Papalymo. After a brief cutscene, your quest will update for you to return to Ailith and report what transpired with Urianger. Returning to Ailith, she states the the information you've provided is of much value but she refuses to leave the forest. However, she does give you an item to bring to Adders' Nest, "Ailith's sealed oath".
Return to Fulke at the Adders' Nest to complete the quest.