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"v=spf1 ip4:xxx.xxx.xxx.xxx include:_spf.google.com -all"
mxmechanism. I'd also try changing
"... the emails sent through the ssh are received with the sender firstname.lastname@example.org. Why not email@example.com?"
mail [your arguments here, such as -s for the subject and the recipient's email address] -- -f firstname.lastname@example.org
mail -s "Test email" email@example.com -- -f firstname.lastname@example.org