Continue to the right. Grapple the golden ring and climb onto the ledge above. (There is a ring in the co-op game, but you could also use the Egyptians' shields for a boost onto the ledge.).
Climbing up triggers the appearance of a Mark of Set, which spawns several enemies.
You may want to drop back down to the walkway so you have room to maneuver as you fight them.
Afterward climb up again and work your way over to the left...
...carefully jumping across the gaps with the spinning gears. You can raise the Staff to alter their speeds, but this is not necessary.
Beyond the spinning gears, head up the stairs to a wall with a series of cycling flame vents. Grapple the ring on the ledge above...
...and wait for the flames to move about halfway up the wall before you begin to climb up.
Ignore the red skull (3/5) for now. Just climb up on the right side of the wall, following the flames.
At the top, wait for the flames to cycle off again...
...then hang from the left side of the ledge to snag the skull. Quickly pull up again before the flames reach you.
In co-op mode, Lara or Carter can grapple the ring to make a tightrope so the other character(s) can climb up, before following them carefully, as in the single-player game.
Retrieve the red skull by hanging from the ledge, or have one character lower another down to the skull using the grapple.
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