It just sounds unnatural. Επιτρέπω sounds a lot more formal when asking for permission, αφήνω is just "let". "Allow me to wake him" is a bit much, right? In any case, you wouldn't hear that in Greek. Also, we say " 'Αφησέ με " αλλά " Επίτρεψέ μου ", at least in the south :) And the pronoun has to be there: επιτρέπω σε κάποιον να κάνει κάτι or του επιτρέπω να κάνει κάτι.