The Fake Powerpuff Girls are actually male criminals committing crimes dressed as the real Powerpuff Girls. They commit the same crimes in order (bank robbery, jewelry theft, and harassment of the Mayor and Ms. Bellum).
In the episode "Powerpuff Bluff," the three criminals held everyone at gunpoint at the bank. The leader of the group ordered the female banker to hand over the money while addressing her as "Sweetcheeks," but the female turns out to be cardboard while Blossom was taped to the back of the cardboard. Then Bubbles and Buttercup bust out of the safe and all three girls defeated the criminals and they were sent to jail. They next rob a jewelry store, but the Powerpuff Girls were hiding in the loot and defeated them again. They later held the Mayor and Ms. Bellum hostage in the Mayor's own home. They were willing to take the Mayor's "Priceless Really Rare One-of-a-Kind Porcelain Poodle," but were defeated by the Powerpuff Girls yet again (and the Mayor gets saved too).
As the three criminals were taken away, the Powerpuff Girls dropped the porcelain poodle (which he didn't mind). Later, they were seen very upset in their jail cell (after watching the news about them), that they decided to not only get revenge on the Powerpuff Girls, but they are willing to get rich. They figured out that they could disguise themselves as the Powerpuff Girls and not only get out of prison, but to frame the girls for the actions done by the criminals. Once they acquire the disguises, they tricked a prison guard into letting them out.
They then enter the bank and acquired the money in a polite manner (which the civilians didn't mind, since they assumed they were the Powerpuff Girls) and acquired the jewelry in a polite manner (the jeweler didn't mind either).
The three criminals were then confronted by the Mayor and Ms. Bellum and were asked to come to the Mayor's home. Once there, they gave the Mayor a minor beating (and one of them gave Ms. Bellum a major clue that the Powerpuff Girls she and the Mayor are currently encountering are phonies by saying "Sweetcheeks"). When Ms. Bellum told the Mayor what is really happening (the bank robbery the second time, the robbery of the jewelry store the second time, and the harassment that occurred just now), which lead to the Mayor foolishly ordering the Chief of the Townsville Police to arrest the Powerpuff Girls, which lead to the real Powerpuff Girls to get arrested at Pokey Oaks Kindergarten.
Once the real Powerpuff Girls are locked up, Ms. Bellum came to tell them that she knows they are innocent and told them about what happened while they were at school. When Ms. Bellum told the girls what one of the criminals said (Sweetcheeks), the girls figured out who was behind everything that got them in trouble. The girls busted through the prison walls and flew to where the three criminals are at. At a Chinese Restaurant, the three criminals were starting to eat their food when the real Powerpuff Girls came into the restaurant. At first, they mistakenly fighting each other until Blossom came up with a idea (each of them fight their own counterpart). One by one, each of the criminals were taken down and their masks were shattered into pieces. They were taken back to prison by the end of the episode (but the real Powerpuff Girls also get some time in jail for breaking out in the first place).