Return to Grace

Walkthrough
Upon first entering Solitude, the Dragonborn witnesses the execution of a man, Roggvir. Speaking to a young girl there named Svari, later, she expresses sadness at never being able to not see her uncle again and mentions that her mother Greta is so upset that she has stopped going to the Temple of the Divines.

Speak to Greta about going to the Temple of Divines
The Dragonborn can then visit Greta at her home, Addvar's House, and hear her side of the story. Greta explains that she cannot continue going to the temple alone, but that if she had her brother's Amulet of Talos, she might be able to go back.

Retrieve Roggvir's Amulet of Talos off of Roggvir
Roggvir's amulet is on his body, which is either still on the execution block, or possibly in the Solitude Catacombs.

Return Roggvir's Amulet of Talos to Greta
Once the amulet has been secured, it simply needs to be returned to Greta.

Bug

 * Clearly Greta rewards the Dragonborn with something, but the HUD does not mention what. Apparently the reward is gold, but if the Dragonborn carries more than one Amulet of Talos this will cause the bug.