Volume 6 first. Cultural festival happened and every class is supposed to nominate 2 representatives to become event organizers for the festival. 8man tried to weasel his way out and took a nap in the infirmary but in his absence he became the male representative (I think it's due to sensei trying to keep him nearby, read between the lines). For the female representative the class argued back and forth, started by a retarded talentless bitch Sagami Minami who despises Hikki+Yui, but then Miura and Hayato stepped up and repelled Sagami, making her the female representative instead. The rest of the classes decides on a musical in the mean time. Yukinon is also a representative as well. During a meeting with the student council, Sagami volunteered to be the overall in charge. She knew she didn't have the talent to do it but to increase her self worth she went ahead anyway.*
*Note: Sagami puts a lot of attention on a person's worth. She classifies people based on their worthiness, and thinks she's very high on the hierarchy. Sadly for her, she ended up in 2-F which had the MiuraHayato alliance who were more popular than her. To put herself on top of those 2, she does stupid shit like things to increase her selfworth or putting people down. In volume 5, she chanced upon Hikki and Yui going out together, and she pretended to be friendly with Yui. According to Hiiki's analysis, Sagami saw only a popular person going out with an unpopular person as "an act of charity", Hikki also noted that Sagami's smile was one that's ridiculing the both of them.
Continuing on, Sagami is also one to surround herself with allies and make cliques (to circlejerk and increase selfworth), so she establish a clique in the organizing committee. Sagami also approached the Charity Club and managed to obtain Yukinon's help despite Yui's objection. Turns out Yukinon's onee-chan Haruno used to be the head of the organizing committee back during her days so Yukinon's motivated to outdo her sister. Sagami couldn' do much shit and told everyone to "relax and enjoy the festival", after which more and more people stopped dropping by the organizing committee to help out, and more work are piled on Yukinon, Hikki and the student council. Haruno and Hayato later came to help out as well.
There's a lot of stuff to read between the lines here, like Haruno's exchange with Sagami, Hayato coming to help out and the student council president Meguri. You can assume those 3 are planning something in the background and it's pretty obvious they're gonna ostracize Sagami because she really is a fucking retard.
Anyway Yukinon collapsed due to the workload so Hikki and Yui went to visit her while others hold the fort. This is where Hikki decides to put in effort to help Yukinon. (He didn't trust Yukinon from volume 4-5 since she never said anything about running him over. In fact, one of Yukinon's favorite sentence is "O don't lie" or something, but she clearly lied about not knowing Hikki even though her car ran him over. Hikki didn't trust Yukinon from then on.)
Shit nearly goes down one day when they committee met to discuss the theme of the cultural festival, long story short Hikki said it should be "Hito (人)", it looks like 1 person is supporting the other, 1 person having fun at the expense of the other. This enrages the whole committee and Hikki became hated. But it worked out in the end since people saw the public enemy no.1 working hard, they wouldn't want to be below him and worked hard as well. For the theme of the festival, they decided on some retarded crap.
Hayato, Meguri were shocked by Hikki's outburst, and got angry at him. You can interpret this as them "getting angry at a dissenter" or "Hikki spoiled our plans to make Sagami a scapegoat, we wanted to install Yukinon as the true committee head".
On the day of the cultural festival, Sagami fumbled on the opening speech since she wasn't prepared nor did anyone prepare her for it. Rest of the festival went quite well though there were problems here and there, but on the second and closing day Sagami disappeared. The organizing committee, as per tradition, requires the head organizer to make a closing speech and announce results of the most popular attraction (which is info only available to Sagami), but no one could find her. Hayato, Meguri, Yukinon and Hikki decided to look for Sagami, Hayato (+Miura), Yukinon and Haruno added in more live performances on the spot, Hikki got the shitty job of looking for Sagami. Hikki enlists Zaimokuza's help and found Sagami on the rooftop. He couldn't convince her , but then Hayato and Sagami's clique came to the rooftop as well. They couldn't convince Sagami to do her job, so Hikki decides on a radical solution. He insults Sagami pretty hard, saying things like "No one was actually looking for you because they want to, I'm a person who doesn't give a shit about you and I managed to find you first. Your friend don't like you at all." Hayato got angry, held 8man by the collar and almost punched him whereas the other girls claimed that 8man was an asshole for saying stupid shit, they managed to get all chummy with Sagami again and Hayato lead them away. 8man said it was all according to plan, he was counting on Hayato's anger to make it even more convincing.
8man went back to the auditorium and saw Yukinon, Yui, Haruno and sensei (playing a bass) performing. Haruno and sensei go way back and she had sensei play the bass back when she was a student. Anyway Sagami did her job and fumbled on her lines yet again, and missing out on a prize category during the closing ceremony, but it was a closure, 8man claims to have helped Yukinon, Yui, the festival and completed Sagami's request (which Yukinon accepted).
Regarding 8man's thought process: He couldn't convince Sagami back in the rooftop so he decided on the spot to realize his "edgy" ideal, which is "The world doesn't need truth, it needs a solution" so he took the self harming path and Hayato played into it. 8man also used objective facts to his advantage, which is him finding Sagami before her clique did.
Back to the festival, everyone decides to go to an after party (organizing committee and the class) but no one invited 8man since the rumor of him trashtalking Sagami twice already gotten out, everyone leaves and Sagami pretended he doesn't exist (it was a terrible attempt since she stopped talking when nearing 8man, 8man noted that this lass' ignoring powers are still green). Student council president Meguri told 8man "thanks for the hard work" (you can read between the lines again, she once said she likes how gentle 8man is, then got angry at 8man, but still is one of the few who talked to 8man at the end, so did she like 8man or not? Interpret it yourself) Someone talked to 8man and asked if he planned everything, then says that 8man saved everyone and the festival, 8man denied and said he had help from Hayato (read: manipulated Hayato, important in vol 7) and the voice said 8man was being humble. Turns out it was Haruno, and Yukinon told her to go home, but Haruno said something along the lines of "I like Hikigaya-kun's heroic ways, it's a waste to pair him with Yukino-chan" (you can take it as it is, but Haruno is one of the character whom you cannot trust at all, most, if not all of her lines have a deeper meaning, also, she's a twofaced person). 8man complimented Yukinon and she blushed, then Haruno and Yukinon went off.
Sensei's turn next.
Sensei tells 8man that he did a magnificent job but became a scapegoat in Sagami's place, and it's not something that should be complimented.
She hold his face with her hands (so 8man can't look away) and said, "Hikigaya, helping others doesn't mean you shouldn't inflict harm on yourself even though you can just shrug it off."
"Well, it didn't hurt me at all."
She holds 8man by the shoulder and says, "...even if you're used to it, but seeing you getting hurt will hurt others as well, you should start being aware of this."
After that they went off, and 8man wanted to complete his report (that's his job in the organizing committee, paperwork) so he went to the Service Club where he can find peace. Turns out Yukinon's there, so they just do their own work quietly.
"Hey Yukinoshita, can you..."
"Sorry, I can't."
"But I didn't even complete my sentence."
"I've said it before. We can never be friends."
"Okay..."
"Yes, I'm not a person who tell lies." (Yukinon's favorite line, also reading between the lines might tell you something, I assume that Yukinon already consider them as friends or even more, that's why the fallout in vol 7 was even worse)
Yui appears later on to interrupt them, and tells them to attend the after party, though both of them rejected that notion, with 8man claiming that he has to work on the report (liar, last line of the volume says he's actually done).
End of vol 6, what happens after is in the drama CD that's bundled with vol 7.