Hilda will divert the knights loyal to her, while you enter the Vault to rescue Aymeric.
Requirements
This quest is available if you have completed the quest Fire and Blood.
Starting the Quest
Talk to Hilda in Ishgard to begin the quest.
Speak with Lucia
Talk to Lucia to proceed.
Speak with the Temple Knight Squire
Use an Aethernet Shard to go to either the Tribunal Aethernet Shard or the Saint Reymanaud's Cathedral Aethernet Shard. Go to where the orange arrow is pointing in the minimap and talk to the Temple Knight Squire.
Special Characters in Duty Support
The Vault can be completed with special characters if you use the Duty Support system.
The Vault
Use the Duty Finder or Duty Support to enter the Vault. Please see the The Vault Guide for a walkthrough.
After you complete the duty in The Vault and exit the area, there will be a cutscene. This one is very important, so it is recommended to watch it either now or via the Unending Journey in an inn room.
Speak with the Temple Knight Squire Again
Talk to the Temple Knight Squire to proceed.
Return to Fortemps Manor
Go south to Fortemps Manor and talk to the House Fortemps Guard to enter.
Speak with Alphinaud
Talk to Alphinaud to complete the quest.
Next Main Scenario Quest
After you complete the quest, talk to Alphinaud to accept the next quest: The Sins of Antiquity.