Q's initial assignment was to determine if Amanda was really a Q. If so, she had to return to the Continuum; if not, she was to be killed. Once he confirmed she was a Q, she was given only two choices - join the Continuum or refrain from using her powers. In neither scenario was there an option to simply take away Amanda's powers. We know the Q can do this since they did it to Q himself in "Deja Q." Since Amanda spent most of the episode considering her new abilities a burden, it would seem that simply removing the powers would make her happier and resolve any concerns the Continuum had.
Despite being on the Enterprise for an internship (to act as part of the crew), Amanda was never given a communicator.
When Amanda first meets Q, Picard is wearing his regular uniform. After the meeting, supposedly just a few minutes afterward, in the captain's ready room, Picard is wearing a different uniform. This has happened before ("Ensign Ro").
Geordi: It was all normal, and then the laws of physics went out the window.
Q: And why shouldn't they? They're so inconvenient.
(Amanda throws Q across the room)
Q: She's being impetuous! She's going to have to start acting like a true Q.
Picard: Unless I'm mistaken, she just did.
(about approaching Amanda)
Picard: We must at least appear to be...
Q: What is it about these squirming little infants that you find so appealing?
Crusher: I'm sure that's beyond your comprehension, Q.
Q: I desperately hope so.
Q: Very impressive the way you contained that explosion. What else have you done?
Amanda: I...I don't understand.
Q: Telekinesis, teleportation... (looks at Picard) ...spontaneous combustion of someone you don't like. That sort of thing.
Q: Jean Luc. Sometimes I think the only reason I come back here is to hear your remarkable speeches.
Q's line about Jean Luc's speeches was not scripted. In fact the writers had such difficulty in coming up with something for him to say that by the time the show was filming John de Lancie dropped the line in on his own.
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