Team Star's Fairy Crew is led by Ortega. Their base can be found in North Province (Area Three).
Defeat the Guard at Ruchbah Squad's Base
Approach the base and you will battle Pokémon Trainer Harrington, who has a level 48 Morgrem and a level 49 Hattrem. You get 2548 Pokédollars for winning.
Enter Ruchbah Squad's Base
If you have defeated the other four bosses of Team Star already, be aware that after you defeat the fifth boss of Team Star, the Academy entrance will be blocked by a Pokémon trainer with high level pokémon.
When you're ready, go to the door of the base and ring the bell. However, you can only bring the first three Pokémon in your party.
About the Star Barrage
Team Star's fighting style is called the Star Barrage. You need to press R to send out multiple Pokémon at a time. You can go back to Clive to heal up your Pokémon. You can also use vending machines to help recover your Pokémons' HP, if you find one. You need to defeat 30 of Team Star's Pokémon within 10 minutes to make Team Star's boss come out to battle you. If you need to exit the challenge early, press Y.
During the countdown, move through the base and approach Team Star Grunts to make them send out their Pokémon. Face the enemy Pokémon and press R to send out your Pokémon against them. Your Pokémon will automatically battle them. They will do better if their type is supereffective against the enemy Pokémon's type. You can press ZR to recall your Pokémon back to you if needed.
Defeat Ortega, Boss of Team Star's Fairy Crew
After your Pokémon defeat 30 of the enemy's Pokémon, Ortega will challenge you to a regular battle, where you can use the six Pokémon in your party.
Ortega has a level 50 Azumarill, a level 51 Dachsbun, a level 50 Wigglytuff, and a level 50 Fairy-type Revavroom (the Ruchbah Starmobile).
Afterward, there will be a cutscene where you learn more about Team Star.
You get Team Star's Fairy Crew Badge, and TM079 Dazzling Gleam. (Press A if nothing is happening.) You'll also get some LP, and more TMs will be unlocked in the TM Machine. You will also receive Pokémon materials that you can use to make TMs.
Explore the Base
After this, you can freely explore the Fairy Crew Base, where you will find various items, including a Full Restore near the lighthouse to the north, a Fairy Tera Shard behind the lighthouse, and a TM119 Energy Ball on top of the lighthouse.
If you want to play the Star Barrage again, you can ring the bell at the front gate of the base.