I'm not a Google employee, but my understanding of their policy is very few people will get approved for fully remote work. Everyone else will need a vaccine.

Where does it say that? I didn't see it mandating that people come back in, but maybe that's some background info not covered in the article.

The article indicated the existing general remote work option is extended to 10/18 and that remote work by permission based on conditions will be available after.

If one assumes that Google cannot compel vaccination as a condition of employment (may be true of COVID vaccines in some jurisdictions), this would imply that “not vaccinated” will be a sufficient condition, in those cases, for remote work permission.

"and that remote work by permission based on conditions will be available after."

Ah, I missed that part.

Can they prove/know that you're vaccinated? One could get the vaccine but claim not to the employer, getting both benefits.