I’ll say right off the bat I don’t have an answer for this. So this is a question not a puzzle.
Is there any group of three letters that can be rearranged to spell six different words?
The best I can do is tea ate eat eta tae aet.
And the last aren’t very good
The most I can find in my dictionary file of 838 three-letter words is 4:
eat
tea
ate
eta
sap
asp
spa
pas
Think of the numbers 123. With those digits you could rearrange them to be:
123
132
213
231
312
321
All that you need are three letters that can be in each position of a 3-letter word and have those be real words.
I’d like to see them, too.
It doesn’t state that the words must be 3-letter words.
eat
tea
ate
eta
at
a
If you can come up with an arrangment of three letters that doesn’t contain three letters I’d like to see it.
That’s not how your original question reads.
“A group of three letters that can be rearranged”. How would one letter or two letters qualify as a group of three letters?
Maybe I wasn’t clear. The six words above are words at least according to some dictionaries.
aet “aged (a specific number of years); at the age of” according to Webster’s New World College Dictionary.
tae is the first word of tae kwon do (It’s also a Scottish word for “to”.)
So if you’ll take those two rare words, you can arrange those three letters into the six possible words.
I’ve seen each of the following used in crossword puzzles (only one definition given for each, as some can be clued in many different ways):
AER – As in the Irish airline, Aer Lingus.
ARE – Unit of metric measure of land area (one-hundredth of a hectare)
EAR – Organ of hearing
ERA – Period of time
RAE – Actress Charlotte ____
REA – Singer Chris ___
Who put whiskey in me tae?
I’ll vote for this list being valid. There’s also Stephen Rea of The Crying Game notoriety.
There is a far more common English word spelled ARE 
(It’s actually been used in several posts in this thread).
Oh no! A tie-fighter shot down my tae.
If we’re playing gotcha, then I will suggest the letters a-c-t, which can be rearranged to spell c-a-t, which qualifies as multiple words since it has many distinct definitions, such as:
1 a mammal (Felis catus)
2 a malicious woman
3 catamaran
4 cat-o’-nine-tails
5 catfish
6 a jazz musician
I don’t think this is what the OP has in mind, however.
Semi-zombie warning (your friend is only mostly dead)…
I wrote a short script to look for a solution. Using only words recognized by Scrabble there is no solution that results in 6 words. AET comes closest with 5. For some reason Scrabble recognizes ZZZ as a word, which comes close but doesn’t fulfill all the requirements.
If anybody has a larger dictionary they can send me or point me to I can run it on those. I wonder if the complete list of Wikipedia entries is available?