I have google apps configured for one of my account and I run it on new Apple Mail. If I flag a mail from my iPhone, it gets flagged on my apple mail. Apple mail has a folder named Flagged where it gets displayed and the same flagged mail appears in the normal Gmail Folder under my Apple Mail. Now, If I go ahead an unflag the mail from my iPhone, it gets un-flagged from normal Gmail folder (as discussed above), but NOT FROM FLAGGED FOLDER of apple mail. Please let me know if I need to edit some settings in my apple mail or anywhere else to make this possible.