Mourning Never Comes

Background
Muiri requests that Alain Dufont be killed, and optionally, that Nilsine Shatter-Shield also be killed. You can get this quest by speaking to Astrid in the Dark Brotherhood Holdout after completing the first three murders.

Talk to Muiri
After Astrid gives you the contract, head to Markarth and speak to Muiri. During shopping hours, she can often be found in The Hag's Cure. She will explain the situation, and give you the various quest markers. If you continue questioning, she will supply two vials of Lotus Extract.

Kill Alain Dufont
Travel to Raldbthar. Once inside, you can either move past the flames to a Master level lock on a gate (Note: This gate seems to be leveled to your character), or work your way around the ruin. This is an unknown theory, either way leads to a platform above Alain but it may depend on the user.
 * Note: If you have already cleared out Raldothar and killed Alain Dufont before starting this quest, the Dragonborn simply informs Muiri that Alain is already dead. She seems disappointed, as his death is something of an anti-climax, but accepts your word and pays you the reward. However, this means the quest ends there and there is no option to kill Nilsine Shatter-Shield, making it impossible to obtain Muiri's Ring.

• Beneath Alain is a small amount of oil. If you use a fire spell you might be able to burn him (and the two other bandits) in one attack.

Kill Nilsine Shatter-Shield (Optional)
There are potential downsides to this objective. Muiri will reward you with Muiri's Ring if you kill Nilsine. A simple way to deal with this is simply travel to Windhelm to murder her while she sleeps in her home.

Talk to Muiri
Head back to Markarth and talk to Muiri to receive your reward.

Bugs

 * If you kill Nilsine and you have not yet purchased Hjerim, the quest Blood on the Ice may not trigger for you as Nilsines mother will have committed suicide and you may be unable to start the quest to purchase the house.
 * There are times at which you might encounter a glitch where you killed both Alain Dufont and Nilsine.You may not be able to turn them in .(PS3 Glitch)
 * Unrelated Glitch: It is impossible to step into the city after completeing The Foresworn Conspiracy due to mineing all of the ore veins in the mine during the quest. It is impossible to serve the sentence, therefore impossible to go into the city to do anything. The option to pay off the bounty does not appear. Although, some believe that the giltch has a fix, this being done by sneaking back into the Shrine of Talos and killing the waiting guards and then the option appears to pay of your bounty.