How to get this quest
You can find this quest in the Adventurer's Guild after completing Rise to Intermediacy and meeting the Arch-Cardinal in Verdeaux.
Skipping the domain
When you talk to Elaine, an option will be available if Aura's Vice is at least 10. Taking this action will allow you to complete report to the guild without any more effort. Doing so makes you forfeit the rewards coming from clearing a demon's domain. As long as you didn't uncover Persephone's domain, the white priestess will allow you to skip it.
To uncover Persephone's domain, interact with the grave in Jacob's farm and spend 25 HP.
Persephone's domain
The domain is pretty straightforward. A 35 AGI jump point allows you to ambush Persephone. A 55 AGI jump point will even allow you to skip all other encounters, but will make you miss a negative energy point necessary to deprive the enemies of a 50% MATK buff. Brittle II is advised for this domain because of the Rigid trait on the Possession fragments.
Persephone is accompanied by two fragments (Possession and Covetous Gaze). Persephone is resistant to PHYS, has 80 DEF and summons a human armor with the Thunderbolt II + Rampage II combo. Starting the fight with Brittle II will help clearing the fragments quickly. Besides Thunderbolt II, you'll take only take DARK damage in the fight. Persephone has 2500 HP and 50 MDEF, so high MATK combined with Light II and Radiance is an option to efficiently damage her. Besides magical damage, you can also use Pierce and Lightning Blade.
Once you've beaten Persephone, report to the guild to complete the quest.