Taisha
The Taisha was founded sometime after the Vertex first appeared, 300 years ago, and started working on the Yuusha System. During the first year after a fire sacrifice bought peace for humanity, the name of the Taisha changed from "Grand Shrine" to "Amnesty" to be reminded to take back the land that was stolen from them one day.
A prominent change in structure came when Miko Uesato Hinata revealed her "Oracle" power, and stated that all Mikos' words were to be taken with the same value as the Shinju, effectively putting her in control of the corrupt priests that made up the organization.
The Taisha of the present are a pseudo-religious organization with political power who spread the worship of the Shinju-sama (Divine Tree). While they practically ruled the island of Shinkoku, where they originated, it has been a slow process setting down roots on the mainland... And yet their efforts have borne fruit, of late, with a small shrine to Shinju-sama in many classrooms, offices, and businesses.
Unknown to the public, they are the sole operators and creators of the Yuusha System, a formalized type of Contract Magic that relies upon the divine blessing of the Shinju-sama to transform young girls into Heroes. The details are even more secret than the fact the Heroes exist to begin with, and no one outside of the Taisha can guess at the intricacies of such a system. They are, on the whole, a force for good, but their objective is solely to prevent the eradication of humanity. To that end, they are willing to use virtually any means, making them not quite the heroic organization they portray themselves as.
The Taisha wear masks and all have the same article of clothing. They are also known to keep secrets even from their Heroes for the sake of worship.
Also see Yuusha System, Yuki Yuna Is A Hero.