Well, it could be because Hermione did something that Annabeth considered a betrayal in the context of the fanfic. Maybe she broke a pact or made a decision that went against Annabeth's plans.
Another reason could be related to a love interest. If they both liked the same character in the fanfic world, and Hermione somehow got an advantage in winning that person's affection, Annabeth, being jealous or feeling wronged, sent the howler. It's a common trope in fanfiction to have relationship - based conflicts that lead to such extreme reactions.
Well, it could be that Draco did something to cross Annabeth in the fanfic's plot. Maybe he messed with one of her plans or said something disrespectful about her friends.
Perhaps Draco has been bullying some of Percy's friends or younger students that Percy feels responsible for. In fanfiction, the relationships can be tweaked a bit. Percy might be more hot - headed in some versions and not put up with Draco's pure - blood supremacist attitude towards those he deems as 'lesser' students. So he sends the Howler as a strong warning.
However, if Hermione knew she was in the wrong, she might feel ashamed. But still, she would likely try to make amends in a more private and rational way. She wouldn't just take the public humiliation lying down. Instead, she'd approach Annabeth later and say something like 'I'm sorry for what I did, but the howler was too much. Let's talk about this properly.'
Well, it depends on the context set by the fanfiction author. It might trigger a conflict or a new plotline. Maybe it reveals something unexpected about Annabeth or changes the dynamics between the characters.
He'd probably be shocked at first. After all, getting a howler is quite embarrassing.
It might be due to cultural differences. Hermione comes from the wizarding world, and Annabeth from the demigod world. In fanfiction, these differences could be exaggerated, and Hermione might find Annabeth's 'alien' culture off - putting or difficult to understand, leading to dislike.
One reason could be their different areas of expertise. Annabeth is highly skilled in architecture and battle strategies in the Percy Jackson world. Hermione, being a brilliant witch in the Harry Potter universe, might feel jealous if she perceives Annabeth as having more practical and hands - on skills. Another factor could be their relationships. If Annabeth has a really great relationship with someone Hermione admires or wishes to be closer to, it could stir up jealousy. For example, if they were in a shared universe and Annabeth was closer to Harry or Ron than Hermione was.
Annabeth is a very determined and strong - willed character. In her mind, Percy is part of her 'territory' because she has fought beside him and for him. She may feel that she has a claim on him, and this sense of ownership makes her territorial.
One possible plot could be that the Howler is a warning about a dangerous magical experiment Hermione is about to undertake. It scolds her for not considering the potential consequences and warns her to stop. This would put Hermione in a difficult position as she has to decide whether to heed the warning or continue with her pursuit of knowledge.
It might be because Snape said or did something hurtful to Lily's friends. Lily is a loyal person, and if Snape insulted or harmed her friends in some way, she would be furious enough to send a Howler. In the context of fanfiction, there could be a new character introduced who Snape treats badly, and this angers Lily.