This is Yuri Nakamura, leader of this faction. She says her enemies are angels, but at first blush the person she's aiming at looks like a normal girl.
Our boy thinks she might not be a good guy.
He rebuffs the recruitment and tries with the alleged angel.
Otonashi is mostly buying time. Amnesia is common among new arrivals, but memories return eventually. He'll wait until he remembers who he is before he really decides to stay with this group.
They prepare their next mission: to steal cafeteria lunch tickets.
The plan goes like this: the distraction squad will put on a rock concert, then, at the height of the concert, they'll activate a giant fan, blowing all the lunch tickets in the cafeteria out where they can grab them.
Too bad she's invincible. When she comes up against him, she can parry his bullets. When the others back him up, she activates a shield that deflects them.
They buy enough time for the operation, but deal basically no damage to Angel.
Ever hear the phrase <better the devil you know>? Naoi, the vice-president of the student council is now acting president, and he's even tougher than Tachibana. Plus he's an NPC, so they really shouldn't go after him.
Otonashi reaches out to Tachibana, but Naoi imprisons them.
He wakes up in the school infirmary.
He's uninjured, but the blood on his clothes tells a different story.
Another boy wielding a halberd comes to visit. And by visit, I mean kill him 100 times for the personal insult of not joining Yuri's group.