The next morning, Hermione finds the article Percy alluded to on the front page. The headline reads that Fudge has appointed Umbridge to the position of "Hogwarts High Inquisitor." It includes comments from Percy himself as well as a discussion of Educational Degree Twenty-two, which gives Fudge power to appoint teachers if Dumbledore can't find one. As High Inquisitor, Umbridge will also be able to inspect the other teachers. Harry and Hermione are livid, though Ron is pleased at the thought of what McGonagall will say when she's inspected.
Umbridge's appointment to High Inquisitor shows the lengths to which Fudge is willing to go to take control of Hogwarts. (Note that the name of this position also calls back to the Spanish Inquisition, a brutal time in history when those who were perceived to be out of line with those in power were tortured and killed.) Now, Umbridge will be able to take issue with teachers' classes and force them to change things to fit Ministry standards—which, given what Sirius said, means that students won't learn practical skills at all. In doing this, Fudge continues to sacrifice the future Wizarding world for his own short-term political gains.
In Potions, Harry's essay receives a D. He tries his best when he brews his potion for the day, and it's at least the right color. At lunch, Hermione, Fred, and George discuss the O.W.L. grading system. The lowest grade is (according to the twins) T for "Troll," and Harry vows to work harder so he doesn't receive T's. The twins also say that Umbridge inspected Flitwick and it went fine. Harry and Ron find Umbridge in Trelawney's tower, ready to conduct her inspection. Trelawney looks disturbed as she sweeps around the room and answers Umbridge's questions. She gives Umbridge a scandalized look when Umbridge asks her to predict something, but then shakily predicts "grave peril." Harry feels bad for Trelawney until she angrily interprets his mundane dreams as foretelling his death.
Harry's ability and willingness to feel sympathy for Trelawney shows that he's growing up and in doing so, is developing empathy for people who are different than he is. Given what the reader knows about Trelawney, predicting Umbridge's death isn't exactly out of character or unexpected—but Umbridge's reaction points to the fact that since Trelawney does this with exhausting regularity and nobody's dead yet, it's not effective or compelling.
In Defense Against the Dark Arts, Umbridge assigns the class to read chapter two of their textbook. Hermione again raises her hand and Umbridge tries to whisper to Hermione privately. Hermione says she's read the whole book already, and she disagrees with the author's insistence that jinxes aren't useful. Umbridge takes five points from Gryffindor, insults all the former Defense Against the Dark Arts teachers except for Professor Quirrell, and then gives Harry another week of detention when he points out that Quirrell was possessed by Voldemort. Harry does his best to not complain when his hand starts bleeding immediately after his first line that night.
Giving Harry another week of detention makes it very clear that Umbridge is willing to physically harm students and do whatever it takes to keep Harry quiet. Further, as the High Inquisitor, it's unlikely that any other teacher would be able to stand up to her at this point, at least if they didn't want that to count against them in their inspection. Harry's desire to speak the truth shows the power of telling his story, which allows him to remember that his experience was real.
The next morning, Angelina yells at Harry for getting detention, and McGonagall chastises her before also scolding Harry. Hermione suggests that McGonagall has a point, and Harry doesn't speak to her until Transfiguration, when they find Umbridge there. Umbridge tries to interrupt McGonagall as she introduces the lesson, but McGonagall points out that she doesn't usually allow interruptions. Umbridge remains in her corner scribbling notes for the rest of class and after class, Harry smiles at McGonagall.
By treating Umbridge like just any other student in her class and demanding respect and silence, McGonagall uses her power as a long-standing, respected teacher to show the students that someone will stand up for them. This kind of rebellion shows Harry too that he doesn't need to go for big actions to make a difference; he can just stand up for what's right when he can.
Umbridge inspects Grubbly-Plank that afternoon and asks questions about Hagrid's absence. Grubbly-Plank refuses to answer, but says that Dumbledore is a supportive boss. Umbridge asks students about injuries in the class, and when Malfoy mentions that he was attacked by a hippogriff, Harry points out that Malfoy didn't follow Hagrid’s directions. Umbridge gives Harry another night of detention.
An ethical inspector would of course want to know that Malfoy was hurt because he didn't follow directions. By punishing Harry here, Umbridge clearly shows that her job is to suppress the truth when it doesn't support her aims of bringing the school under her control.
That night, Harry leaves Umbridge's office just before midnight. His hand bleeds through a scarf and Hermione offers him a bowl of pickled murtlap tentacles to soak his hand. Harry refuses to complain to McGonagall, but Hermione says they have to do something about Umbridge. She suggests that they should learn Defense Against the Dark Arts themselves, as it's important to prepare themselves. She suggests that Harry teach them, and Ron seems intrigued by the idea. Harry thinks they're joking, but Ron lists Harry's accomplishments. Harry snaps and shouts that it's not all memorizing spells; fighting Voldemort is terrifying. Hermione says they need to know what that's like.
Hermione's suggestion shows how seriously she takes both her O.W.L.s and the threat of Voldemort's return. She also has a point that even in a safe classroom setting, it's difficult to prepare for the realities of fighting someone who's fighting to kill. Because he's fought Voldemort before, Harry's lived experience is a valuable thing for Hermione and others to hear about and learn from, as listening to him will help them be prepared for what's to come and understand what this kind of fighting is like.