What should I do with amulet of Talos?

Solution 1:

This quest has multiple bugs.

To complete it, all you have to do is return the amulet to Greta.

If you've done that, and it's not complete, you've probably encountered one or both of these bugs, taken from the linked wiki page's Bugs section:

• If Roggvir's Amulet of Talos is taken before this quest, any Amulet of Talos picked up after that will be marked as a quest item and cannot be dropped or sold.
• This quest may stay in the miscellaneous quest section even after completion.

The reward for completing the quest is just a leveled pile of gold, so it's not something to really worry about, even if having it in your quest list is ugly and bothersome.