You look for a maximal set of ordered four-tuples that each consist of either four identical or four completely different letters

The OP said "same or sequential alphabetic neighbours" and later said the alphabet is ADEFGMSTV. He also clarified that order isn't important.

My best guess is that the following are allowed

• AAAA
• ADEF
• AAAD

and the following aren't

• AEFG (A doesn't neighbour E, F or G)

Please confirm.

What is the exact rule of how one tuple can "destroy" a solution, if overlap is allowed anyway?

A given instance of a letter can be reused. He's talking about overlaps in the input string.

  +-+-+-+----- tuple 1 matches character between indexes 0 and 6
  | | | |
  AADDEEFF           overlap over indexes 1 to 6
   | | | |
   +-+-+-+---- tuple 2 matches character between indexes 1 and 7
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