So the student said that the key must be F major because the starting note was F. But then I asked what does the C sharp next to a few notes imply? He said, it must 5th raised scale degree.
Strange, isn't it? How often do you see one of the constant scale degrees (1st, 3rd, or 5th) altered?
There is a simpler explanation, though.
You see, although the exercise started on F, it ended on D. But D is not a constant scale degree in F major. It's of course the 1st scale degree in D minor, though.
And C# in D minor means raised 7th scale degree which is a sign for harmonic minor.
That's how you can identify a key in your organ piece - look not only at the key signature, not only at the starting note or chord, but also on the ending note or chord and check for alterations.
By the way, here are the signs of the altered minor mode: raised 4th and 7th, and lowered 4th and 2nd scale degrees. Altered major has raised 2nd and 4th, as well as lowered 6th and 2nd scale degrees.