As soon as the preceding quest is completed, you’ll be starting off in the beach facing the island of Ithaka. Hop on the boat and sail towards the island. Before disembarking in the island however, you’ll pass by an optional location Ktesipos’s Shipwreck. Have Ikaros scout from the air to tag the treasure chest submerged underwater. There’s no shark so just dive down and retrieve the loot to complete this location.
Continue to the island and head towards the ruins of Odysseus’s Palace. There are a lot of enemies in this ruins so make sure to tag them all. Additionally, you can find ancient tablets and an Ainigmata Ostraka in this location as well. Prioritize whittling down enemy numbers to make your life easier.
The shroud is located inside the main ruins, held by a captain. There’s always a lookout beside him but you can take him out without alerting the captain. First, work on taking out the guards in the immediate area and hide their bodies.
Then, while the lookout is looking away, you can approach beside the wall and assassinate him when in range so your character pulls him behind the wall and finishes him off. From there, you can now sneak behind the captain and perform a sneak attack to deal massive damage. This will not kill him instantly but as long as you take out the lookouts near the building, nobody will be alerted of your presence.
Hide their bodies as well then loot the shroud. Technically, you’re done with the task and you can return to Elpenor to complete the quest. However, I will still recommend scouring the area to get the Ainigmata Ostraka and the tablet in this area.
There’s also a captive character, Odessa, that you can rescue to start another sidequest (A Small Odyssey) but I recommend releasing her only after taking out all enemies in the area.
Once ready, head back to the Temple of Zeus near Sami and find Elpenor. After the cutscene, the quest will be complete. You’ll also get the rare head gear, Penelope’s Shroud. You can wear it if you don’t have any other better equipment at this point.