In some fanfics, it could be an entirely new character. This new character could be an old, reclusive wizard who has been observing Harry for a long time. He sees great potential in Harry and decides to come out of hiding to guide him through a unique training that combines different forms of magic that are not taught at Hogwarts.
It could be Dumbledore. He's always been a guiding figure in Harry's life, so in the fanfic, he might continue to mentor Harry through his special training, teaching him ancient and powerful magic.
In some fanfics, Harry might go through combat training with new spells and dueling techniques. This could include advanced defensive magic like creating more complex Patronuses.
Often, the main character would be a young witch or wizard who is eager to become an auror. They might be a new student at the training academy.
It could be Sirius Black. Sirius was a very powerful wizard and he had a special bond with Harry. He could have come back in some way to train Harry in some unique magical skills, like how to transform into a more powerful animagus form.
I would assume Harry Potter is a main character. But depending on the plot of this fanfic, other characters could also be in the spotlight. For example, Neville Longbottom might be a key character if the story is about the resistance against the dark forces at Hogwarts. Or perhaps Luna Lovegood could be involved if the fanfic has something to do with the more mysterious aspects of the wizarding world that she is often associated with.
It could be a former Death Eater who has had a change of heart. This person might see the error of their ways and decides to secretly train Harry, believing that he is the only one who can truly defeat Voldemort. They could teach Harry some of the Dark Arts that Voldemort uses, so that Harry can better understand and counter them.
Well, perhaps in this fanfic, Harry Potter could be receiving special training in advanced defensive magic from a reclusive and powerful wizard. This training might include learning spells that are not taught at Hogwarts, like some ancient protection charms.
Harry would become more disciplined. His natural leadership skills would be enhanced by the military training, making him even more of a force to be reckoned with.
One interesting plot could be the main character facing a series of extremely difficult and dangerous dark magic challenges during the auror training. Maybe they have to deal with a powerful dark wizard who has infiltrated the training grounds to disrupt the training process.
It could be Zeus. He is a powerful god and might see Harry as a worthy candidate for training. He could teach Harry about leadership and power control.
Dumbledore could be the instructor. He might have a special training program in mind for the students. He could teach them about the deeper mysteries of magic, like the secrets of the Hallows or the true nature of the Deathly Hallows. His wisdom and experience would make him an excellent teacher for the students during the summer training.