2BA but unlikely or 3/16" BSC
Both under 5mm, his hole is over 5mm. 2BA is close on pitch, I'm sure we've all put 2BA screws into M5 threads and barely noticed, BSC would be 26TPI which is a long way off, you'd only get an M5 screw about 1 turn into that.
@gbj_tester started
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2BA but unlikely or 3/16" BSC