New primary objective: Kill Lurbuk
Note! This quest is optional and a chance to get it will show up during completing the quest Whispers in the Dark.
In order to unlock this quest, you have to ask Nazir for new contracts during conversation with him. It is worth noting, that in order to unlock next main Dark Brotherhood quest, The Silence Has Been Broken, you have to complete at least one of the side contracts (I suggest completing all of them). Leave the Dark Brotherhood's Sanctuary and open the world map. Lurbuk can be found in the settlement named Morthal (screen above), located in the northern part of Skyrim. Travel there now.
Lurbuk is a bard in the Moorside Inn, located in the northwestern part of the settlement (screen above). Since it is very unlikely that he will come out form the inn, you have to go inside.
I do not recommend attacking Lurbuk while he stays in the main inn room, because you'll almost certainly call attention of local guards. You should wait until Lurbuk goes to sleep to one of the inn rooms instead. Crouch and carefully enter the room with sleeping bard. It is worth using the complete Shrouded Armor, which you've received from Astrid at the end of the quest With Friends Like These (or other, better parts if you have them). Carefully approach Lurbuk (screen above) and get rid of him in any way, stabbing him with a dagger for example or shooting an arrow in his head. After successful action you should leave he inn, because Lurbuk corpse might be noticed by one of guests and the room where he was sleeping has no door.
New primary objective: Report back to Nazir
You can now return to the Dark Brotherhood's Sanctuary and report Nazir what you've done. If you're going to complete other contracts received from him, you should return to the assassins' hideout only after completing them all. Nazir will reward you with gold.