Well, please read that points guy link from December about someone rejected by United flying on expired green card plus extension letter.
What I'm seeing from this thread is that some people flying on some airlines have been permitted to visit Mexico with an expired green card and extension letter, and some have not.
The website of the consulate of Mexico in Houston says:
EAD cards, I-797 A Forms, Advance Parole Forms, I-20, F-1, TPS, I-512, or DS-2019 are NOT ACCEPTED to enter Mexico.
https://consulmex.sre.gob.mx/houston/index.php/visas/visas-english