So I did some experimenting with alternate fingerings and, while the actual fingering for the second octave F# is half-thumb, 1, 2, 3, 5 (not … 5, 6 as in the lower octave), the 5, 6 fingering that I suspect you have been using will indeed produce a slightly sharp E. Therefore, I suggest that you check your fingering chart where you will discover that several of the fingerings in the second octave are different from those in the lower octave. I hope that this is clear and that you find it helpful.
–Dick–