If I understand you, this seems impossible with any number of groups.

In each grouping, one letter is paired with 2 other letters. But for each letter, there are 5 additional letters to be paired with. 2 groups will result in pairings with 4 letters (but not a fifth). 3 groups will result in a duplicate (there are 6 "slots" to pair the letter with, but only 5 letter to put into the pairs).

You could have ABC and ADE, but how will you pair A and F now? If you create a new set AFx where x is another letter, which letter is x? No matter what this will result in duplication.