Officials from some of the biggest LGBT-friendly groups in the country are outraged by the Prez's son's appearance Tuesday on "Fox & Friends" -- when he responded to an NYT op-ed told from the POV of a lesbian woman ... who said she was a secret Trump supporter.

In his response, Eric seemingly flubbed ... making it seem like he's a member of the community. More importantly, he suggested gay people were down with MAGA ... but that's hardly the case ... so say these orgs.

Reps for GLAAD, the Human Rights Campaign, PFLAG and the NTCE Action Fund tell TMZ there's a consensus -- folks in their community are mostly NOT Trump fans, nor do they consider him/his fam genuine allies in their fight for equality and representation.

The facts are clear says the LGBTQ honchos ... the Trump administration has actively worked against LGBT causes -- from blocking trans soldiers from serving to sanctioning discrimination and/or firings of gay workers and couples looking to foster.

Bottom line ... all of these orgs believe Eric is either exaggerating or more likely flat-out wrong in thinking the LGBT crowd's got his pop's back -- especially in this election.