After the cutscenes where the girls ask for help in increasing Planeptune's shares, you will gain the following: Neptune's Violet series, Noire's Piano Black series, Vert's Round-X series, and Blanc's W11 series.
In order to advance the story you will need to accept the quest [Isolation] The League and head for Ario Plateau in Lowee.
Ario Plateau: Arfoire's Status
Poor Arfoire has been relegated to the bottom-rung tasks as a result of her brainwashing by Plutia.
This place is confusing due to all the teleporters and strange layout. Start by hoping up on the grassy platforms on the right and taking this to a higher plateau. There is a treasure with Healing Drink x2 near the flag and teleporter that you should take. There is two treasures straight ahead with a Life Fragment and P. SP Charger. Following the path should lead you to a treasure with a Healing Drink before you reach the area with the Plaid Dolphin and a teleporter. Grab the Puzzle/Logic x3 and then take the teleporter. There are a total of 3 teleporters, you want to take the south one. Save and approach the objective crystal.
The biggest threat is the Renegade creature that likes to hit for 4k damage. Focus on taking out any enemies that are close together that more than one enemy can be hit by one person. This is usually the Dogoo and creature in the back. Once you have cleaned out a group, finish on whatever is left and make sure to keep the girls healthy.
Planeptune: Arfoire Leaves?
Return to the Basilicom where the grown up kiddies will have an embarrassing convo with the CPU's. It appears that Eden has been making Histy have to work overtime and the girls all decide to help.
Eating Compa's Snacks
In order to advance the story you will need to accept and complete the quest [Fierce] Flame Wolf.
After completing the quest and turning it in, return to the Basilicom to not only see IF get kidnapped, but the girls learning about the kidnapping...apparently Arfoire really hasn't learned her lesson and Plutie is starting to scare me.
Otori Cave: As Thought
You have been in this exact cave before, as you probably have noticed that Neptunia has limited locations. Start by heading north and then up the ramp and back around to the south, getting the Royal Bracelet (PP). Continue until you reach the split and then take the northern middle path all the way across the bridge where the Metal Dogoo (PP) and Royal Armlet (PP) are. Return to the split and take the western path now all the way north to the teleporter and then head west to reach an upper ledge with the Advanced Blade (PP) and Engage Ring (PP). Now drop off to the west and use the save before reaching the objective crystal.
Pirachu is pretty tough for a little rat and depending on your level, who he attacks, and whether he crits or not you can easily die in one hit. Focus on keeping everyone at 6k+ health with Plutia or Nepgear (Plutia's heal is stronger) and then focus on breaking his guard. Once broken, a EXE Drive from almost anyone should finish him off.
Back at the Basilicom, Pirachu will spill the beans on how Yellow Heart is using energy from another place to make her more powerful...I guess we will just have to destroy it.
Anonydeath's Lab: The Creeper's Lab...
Head east into the large room with the mini-boss and flag and grab the Whoaaa! x3 and Black Clover (PP) in the southeast corner. Return to the previous area to the west and head south, getting the Energy Lump (PP) on the east side of the area. now continue south and head for the exit to the east, but make sure to get the two treasures above the zone with Energy Fragment x3 and a Taboo Staff.
Start by heading all the way east to the wall and then follow it south to a treasure with Eject Button x2. Continue to follow the wall south and then east into an alley where three treasures lie: a Life Ring, STR Booster x2, and the AMN-Noire (PP). Now head west across the area and use the save. There are two treasures you will want to grab before heading to the objective crystal, the Cube Hammer (PP) directly west of the save and the Healing Drink x2 in the bottom right of the area. You can then save again, if you want, and approach the objective crystal.
It looks like Mr. Badd's kids don't want him to fight anymore, so Anonydeath is on his own. Poor guy knows he's in for a beating, but that doesn't mean he isn't going to put up a fight.
Boss: Anonydeath (2)
Anonydeath is only dangerous if you let yourself get low on health. Using the usual tactic of healing while breaking his guard and then using your EXE Drive will work fine. Make sure to keep at least 1 EXE Drive meter for the next fight.
Although Anonydeath is down, the girls are too busy to finish him off while he stalls and makes time for Yellow Heart to show up.
Boss: Yellow Heart (2)
Debuffing Yellow Heart's STR is going to be key in this battle, otherwise she will one-shot you. Focus on keeping her debuffed with Neptune and Plutia and using Nepgear, Neptune, and whoever the third party member is to beat her down. When she gets about half health you can go all out with skills to finish her off.
After the obvious "twist" is revealed that Peashy is Yellow Heart, the chapter will end and you will get your evaluation.