I assume that when she released the flyers that said there was a doppelganger at the school, she only knew for certain that there was one - herself - and she was just hoping to flush out other ones. And that’s why she was sort of both surprised and relieved when Nao revealed herself.

I would guess that she’s in a very desperate situation. All indications (particularly Nao’s non-death) imply that the replica is effectively created by the original, since they not only can be summoned or banished at will, but can be summoned intact and unharmed after being “killed.” So the replica is entirely a function of the original, and that in turn implies that the replica cannot exist without the original.