During season 1, Yata and Fushimi have mini confrontations. I learned later in the season that they only hate each other because Fushimi joined Homra's most hated clan, Septer 4. Why does he do this?
I am not sure if this can help you, but I can at least give you my opinion from what I read in K -Lost Small World-. This may be a long explanation so if you go to the bottom, you can see my short-term answers.
Before Fushimi and Yata, joined HOMRA, it was their own small world; both were really close and happy. Until they joined HOMRA. Yata started to have his own world with other people, that it had lesser and lesser time with Fushimi. Lesser and lesser until Yata wasn't Fushimi's light anymore. One time, the blue king asked for a meeting with them; and one of the representatives was Fushimi which Munakata took notice of. Though Munakata did show that he wants Fushimi to work for him, Fushimi didn't agree just yet, though he did ask if it is possible to change clans.
In the manga that I read, it was only until chapter 11, but feel free to check if it's been updated. The chapters 12 to 17 are just story strips. It ended with Munakata telling the green king to pull back from attacking Fushimi.
So I have a few answers:
- He betrayed HOMRA to get Yata's attention. (Cause of Yata)
- He went to SCEPTER 4 because Munakata gave him a choice. (Cause of Munakata)
- He went to SCEPTER 4 because he knows he can be of use to the society and doesn't want to end up like his father. (Cause of Nikki)
- He went to SCEPTER 4 because he thinks that he belongs there, and is not fit to be called a Red Clansman. (Cause of Mikoto)
This is my opinion: He did it for him, SCEPTER 4's, HOMRA's, and for other people's own good. If he stayed in HOMRA, nothing will progress for him. Yes, he is a skilled fighter. But if his brain is put to waste I think he can feel himself becoming like Nikki. So that helps himself, SCEPTER 4, and other people. Now, what about HOMRA? What I like about him going to another group was that he could be a bridge that can lead HOMRA and SCEPTER 4 to an alliance.