On the Loose

At the start of Act III, you are requested by Meredith at the Templar Hall in the Gallows to track down 3 dangerous apostates (Huon, Evelina, and Emile de Launcet) and bring them back dead or alive.

Acquisition
Speak to Knight-Commander Meredith at the Templar Hall in the Gallows.

Walkthrough
Head to the Templar Hall through the Gallows and talk to the Knight-Commander. Afterwards, you can talk to her assistant, Elsa, for information pertaining to the fugitives. Afterwards, you are free to chase them down.(There is "The enigma of Kirkwall" in a room).

To track down Huon, head to the Kirkwall Alienage in Lowtown and speak with his wife, Nyssa. She will tell you that he will be returning the next night. Go into Merill's house after doing this and then back to Lowtown at night to have him appear. Confront Huon, who will kill Nyssa with blood magic and attack you afterwards. Kill him. Upon death, he will drop a piece of your champion armor set [confirmed - It will always be the boots]

To track down Evelina, you just head to Darktown and talk to her siblings, Walter and Cricket. Enter the Sewer Passage afterwards fighting through two waves of demons and shades and you'll come across a trap. There is a switch you can discover as a rogue to deactivate the spike traps. Anyways, you will find Evelina, who is actually an abomination. Kill her. [confirmed - She drops nothing, though there is a Rune of Devastation design in a nearby chest.]

To track down Emile de Launcet, talk to his parents in Hightown at night. You'll learn his mother gave him money to escape, but he is wasting it on alcohol instead and will be found at The Hanged Man during the day. Once you confront him, you have the option to turn him in or let him go. If Isabela is with you, she can provide some comedy relief. Anders also has extensive commentary to contribute, and other party members also have minor comments. Isabela and Aveline and Anders and Fenris have paired comments to contribute as well. Letting Emile go results in,.

Note: If you decide to let Emile de Launcet escape, a letter with 5 would arrive at your desk later in the game.

Afterwards, return to Knight-Commander Meredith to claim your reward and finish the quest.

Result
The Apostates are apprehended and Knight-Commander Meredith is pleased, slightly depending on how you handled Emile de Launcet.

If you place blame of the situation with the Templars when talking with Meredith at the end of the quest, it results in: ,

Rewards

 * 1200 Experience
 * Boots of the Champion (Mage), Boots of the Champion (Rogue) or Boots of the Champion (Warrior)
 * 5, if Emile was allowed to escape - Emile's father
 * You receive also a letter from Meredith with 5

Humorous side note
The De Launcet Estate has the exact same layout as the The Blooming Rose.