For Than’s loyalty mission, you will have to help a talented assassin prevent his son from making the same mistakes in life as he did. The mission takes place at the Citadel, so travel there to talk to Than and get the mission.
Upon arrival, head to the C-Sec offices next to the docking area and talk to Captain Bailey. Ask Than’s son and Bailey will tell you where to go. Head to Zakera Ward Level 28 and then go to the Comm Terminal area. Here you will find a Mouse that will tell you what you need to do. Then return to Captain Bailey, who will arrange the interrogation of Elias Kelham.
At this point, you will need to decide if you are a good cop or a bad cop. You should choose the one that suits your playing morals, so Paragons should be good cops and Renegades should be bad cops. This will take you to the information you need and to the next part of the mission.
In the next part, you will have to track the target, keep them in sight and update Thane at their position. If you lose sight of a person, it will be considered a failure and Thane will not become faithful. There’s nothing wrong with getting a little save-scumming after this part.
Just follow Turian and update Thane to his position every chance you get. Keep an eye on Turian until you reach the point where you will have to use the Apostate or Paragon action to prevent Than’s son from killing him. This is another point where you can fail the mission overall and lose Thane’s loyalty.