Izumiko is upset that Takayanagi and his gang are trying to control her. All the electronic devices in the school are out of order because of her power and thus the school festival, competitions couldn't go on. Takayanagi is also turned into a cute white dog by Izumiko (poor him. I'm sure he would later be humiliated by the Soudas for the rest of his life because of this incident). Izumiko then went to another dimension/layer and wished that she would never go back to the reality again because she doesn't want to be captured and control by anybody. She also realised that she is indeed not NORMAL and couldn't live a normal life with normal people. Masumi appears and persuades her to stay in this dimension/layer forever with him and other ghosts. Miyuki later finds out about Izumiko's disappearance. He then asks Wamiya for help. At first Wamiya rejects saying that Izumiko said 'Don't come' (you already watched this in ep 11) and now he senses that she doesn't want to come back plus she is with Masumi who is apparently stronger than him (if you're reading the novel you could feel ever since the volume 3-4 that Wamiya is jealous of Masumi). However, Wamiya finally agrees only to transfer both Miyuki and Takayanagi the dog into another dimension but they must find a way to Izumiko by themselves.
The climax is how Izumiko comes back to her sense (thanks to Miyuki and the cellphones) and dances to send all the ghosts back. Her dance bewitches the whole school and so makes her the best candidate of the World Heritage. - End of Vol 5-
-Vol 6- is actually how all the things are resolved.
Takayanagi (now back to human form) disagrees with the result and wants Hodaka (the kabuki guy) to reconsider the decision because of his blah blah blah reasons like Izumiko is incapable of controlling her power. So we have a showdown between the two for a final decision. As you can guess, Izumiko wins BUT she declines to accept this candidacy. She asks Hodaka and the student council to announce Takayanagi as a top student/best candidacy for the World Heritage to the school's president.
Next highlight is at the School's Christmas Event in which not only students but parents also joining. This includes Angelica's father who appears to be a person from IUCN. By the way, Angelica is the white girl we're seeing with Takayanagi in ep 11. Another white guy is a German named Claus. Both Angelica and Claus are also people with special powers.
In short, Izumiko (under the reindeer suit) goes panic (for a stupid reason I would say) and chaos happens (there is illusion Izumiko created though not all people could sense it). Miyuki, Takayanagi, the Soudas, other student committees try to cover for Izumiko and thus use their powers to turn things back to normal. Angelica's father impresses with this and saying that he's thinking of making the whole team (later called Team Himegami) to be the World Heritage candidate because everyone in the team deserve to be protected. (In other words, just like the world heritage site, they could announce the 'whole area' being a world heritage instead of just making one specific temple or one particular building to be the world heritage).
The final major event of this volume is when Izumiko is abducted by a doctor/researcher named Nakayama Mizuho. Now this will confuse people who don't read the novel because the anime rarely mentions anything about her. She is Yukariko's friend since college, and is the one who is regularly making a physical checkup for Izumiko ever since she was young. Actually we caught a glimpse of her in episode 1: she is the person who's waving a hand to Izumiko while Izumiko's bowing to her and was about to leave by a helicopter. .......then Miyuki (of course, who else?) comes to rescue Izumiko. There is a sweet scene here when Miyuki tells Izumiko to run and don't worry about him while he's about to confront with the bodyguards. Izumiko ignores Miyuki's suggestion and hugs Miyuki from the back. She says she would rather be with him whether be in a foreign land or any place. Finally, Yukimasa appears and rescue the children and takes control of the situation.
Yukimasa takes Izumiko to meet with Yukariko and then mother and daughter go back to Tamakura Jinja. All about the Himegami things is explained here.
THAT final scene: Izumiko goes out for a walk, enjoying the starry night. Miyuki suddenly comes to the scene which makes Izumiko surprises because she thought he's in Tokyo. Then there are: Sweet talks. Holding hands. And a KISS (?)
