This page of the guide to The Witcher 3: Wild Hunt includes a walkthrough of all side quests related to the Fyke Isle location . In the walkthrough we describe how to solve the puzzle with lighting the fire while searching for the magic lamp, and how to answer the invitation from the witch Keira Metz.
This quest can be unlocked during the main quest Wandering in the Dark - the one in which you investigate the mage hideout accompanied by Keira Metz (M10,6). At the end of that quest Keira will ask Geralt to help her with finding the magic lamp. Accept helping her before she decides to leave the location.
Walk with Keira to the room where you must complete a rather simple puzzle (the picture above). In order to complete it, you must ignite the fire in the correct order: counting from left, you need to ignite baskets 3, 2, 4 and 1.
You can go further. Walk to the tombstone where Keira will find the lamp. Now all that's left is to leave the mage's hideout. Return to the fork where you started this quest and pick the stairs leading outside (remove another illusion at the end of the march). Listen to the conversation with the witch (100 experience points). She will invite Geralt to her and you will receive a side quest An Invitation from Keira Metz.
An Invitation from Keira Metz
This quest is unlocked after you finish the side quest Magic Lamp. It is given by Keira Metz after you find the named artifact in the same ruins where you completed the main quest Wandering in the Dark.
You must return to the witch's hut. She lives in the forest near Midcopse village in Velen. During the conversation, offer your help to the witch. Keira will give you new objects, the Magic Lamp and xenovox that can be used for ranged communication. Now you can start the side quest A Towerful of Mice.