HOW TO CLEAR ALICE'S REQUEST AND GET HER OUTFIT DESIGN
First of all you need to accept her Request in the Lobby, she'll ask you to kill a Behemoth MT-13 in the Lost Gate. Teleport to "Lost Gate Depths", from the Starting point turn yourself 180 degrees and check the teleport hidden on the right (to activate to teleport you need to clear the LDC's main story and accept Alice's request).
I personally found easier to keep attacking his head instead of his body to try and stun him, I recommend to bring 2 healers since it's going to be a long fight.
To craft the outfit we need:
10 Premium Cloth
10 Old Ultralight Armor
10 Shiny Membrane
5 Monster Ligament
We can buy the first 3 from the Shady Merchant Shop as usual on Extreme Mode, the Monster Ligament is dropped by the Shady Delinquent in the Lost Gate Dungeon.
For a video walkthrough check the video below: