E = E G# B
F = F A C
G = G B D
The G sharp is a real oddity there, I don't think a diatonic approach would be the best there. The first key with a sharp G is A (3 sharps) , next closest would be D lydian (3 sharps, D is 2 sharps.
You could try to play G mixolydian (same notes as C major) for a diatonic progression and dodge away from playing G when on E (as a G# would likely throw the feel, it might work though now and then). This might work if your progression resolves to the G.
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