Saruhiko was swayed over to Scepter 4 by his king and Captain Reisi Munakata. Although the decision to switch sides was unplanned it occurred because Saruhiko was unable to find a home for himself at Homra. Scepter 4 on the other hand, acts more like a workplace than a family. Saruhiko is able to know his place because Munakata makes it clear what he expects of Saruhiko. He doesn't have to worry if being a member was a mistake because Munakata made it clear from the moment he met Saruhiko what he wanted him for. As such their relationship is more successful than that of Saruhiko and Mikoto because Munakata is the first one to understand Saruhiko in ways that he doesn't seem to understand himself.
Munakata wanted to rebuild Scepter 4 with the best individuals possible. He didn't rely on the members that already existed and in many cases chosen his own based on their usefulness. He set out to recruit Saruhiko despite the fact that he was from another clan because he needed his abilities with electronics and someone more subtle. Saruhiko as a genius hidden weapons user was ideal.
"He does not do something as ambiguous as believing instead he knows how far I can go. What results I can bring." - Saruhiko Fushimi, K: Days of Blue Ch.4 Pg. 25
He made it clear that he wanted Saruhiko so that Saruhiko would know his value to Munakata. Although he is awkward Munakata understands the fears that Saruhiko had. He likely spent months obtaining information on his past and knew exactly how to approach Saruhiko without alienating him. He pushes Saruhiko into attending team building exercises without overexerting his power in a way that will frighten Saruhiko off.
Ultimately I think Munakata's presence serves as a learning tool for Saruhiko who still lacks many of the social skills that come naturally to adults. Although it is never said Munakata acts as a contrast to Saruhiko's abusive father, Niki. Munakata, like Saruhiko and his father is a genius. Unlike Saruhiko's father however he makes something of himself. I think that while he would never admit it and perhaps doesn't realize it, it gives Saruhiko hope that he too can reach his goals of becoming something more.