Fujimura-sensei is a mysterious English teacher. It can be said that both in the past and present she is the symbol of daily life.
She is a high level AI, [and as such is supposed to manage some part of the tournament], but it is not clear what she actually manages.
In EXTRA she just abuses the protagonist as a gofer (despite the protagonist being completely bewildered at being suddenly thrown into a death game), and in CCC (where the protagonist is in an escape game) they toy with the protagonist as a gofer.
Okay, so there is no difference regardless of the situation.
If you clear all the tricky problems that Fujimura-sensei throws at you, you get access to the satisfying interior of her My Room. But what’s with that picture you take with her?
In EXTRA, if you clear all the Taiga Quests from the second week onwards, Taiga will get cocky and end up summoning something that shouldn’t be summoned. I think that’s quite enough, Fujimura.

Fate/EXTRA Material: Encyclopedia of Fate/EXTRA