After "Brady Bunch" dad ROBERT REED died in 1992 and was posthumously outed as a gay man who was HIV-positive, people started getting familiar with the idea that America's most perfect family was actually a FAKE family.

Still, revelations about their REAL, off-screen lives can be kind of shocking to people of a "certain age".  This is one of them:

FLORENCE HENDERSON, who played perfect mom Carol Brady, has a new book coming out in September... in which she reveals that she got CRABS in the late 1960s, from New York Mayor JOHN LINDSAY.

Oh, and she was married at the time.

It happened at the Beverly Hills Hotel.  Florence says, quote, "I was lonely.  I knew it wasn't the right thing to do.  So, what did I do?  I did it."

Apparently, this was a wham-bam-thank-you-ma'am situation, because Florence and Mayor Lindsay didn't even spend the night together.

Florence went home, then woke up the next morning with, quote, "little black things" crawling all over her bed and body.  She made a quick trip to the doctor to get it taken care of.

Lindsay tried to smooth things over with flowers and an apology letter.  But it didn't work.

Florence says, quote, "Guess I learned the hard way that crabs do not discriminate but cross over all socioeconomic strata.  He must have had quite the active life.  What a way to put the kibosh on a relationship."

Lindsay was mayor from 1966 to 1973.  He died in 2000.

Florence... who's 77 years old... remained married to husband Ira Bernstein until their divorce in 1985.  It's not clear if she ever told him about this particular escapade.

"The Brady Bunch" ran from 1969 to 1974.

Elsewhere in the book, Florence shoots down the often-repeated rumor that she had a relationship with... or even just went on a date with... her onscreen son BARRY "GREG BRADY" WILLIAMS.

She says, quote, "Barry did have a serious crush on me, which I understood and helped him get past.  Let us just say that if he had entertained a roll in the hay with me, I would never have done it."

But she adds that she and Barry remain good friends to this day.  (Florence's book, "Life Is Not a Stage", comes out in September.)