Well, my head is just over 7mm, so it falls between the gaps with what you've discovered, big-all.
However, denso13, your system seems to suggest the screw could be a no 7 (as big-all suspected). It's nearly 4.5 16ths, so:
(4½ - 1) x 2 = 7
But no 7 shanks are massive compared to the...