Groups with the most longevity, with few personnel changes, and still manage to be popular -
The Stones, Pink Floyd and Aerosmith
Most influential group of the 80's forward -
Roxy Music
Most innovative group ever -
The Beatles
Most influential 'hard' rock group of the 60s -
while The Beatles were the most innovative, the most influential group of the early 'hard' rock genre was The Kinks, and I would say they were followed closely by The Animals. Most music fans were being blown away by The Beatles, but most rock musicians were being blown away by the latter two groups.
Best Prog Rock group -
King Crimson, at least at its conception, with its outstanding The Court Of The Crimson King. It took the rest of the prog groups a couple of albums to even begin to catch up to what the group achieved with that album. And I say that because Pink Floyd was being labeled at the time as a Psychedelic, or Acid, Rock group, but even so, although Meddle was a very good album, it was a couple of years later, and Dark Side Of The Moon was still about 4 years after when Court of the CK was released in '69.
Interesting trivia note that some early rockers may know. When Greg Lake and Carl Palmer left King Crimson and were forming a new band with Keith Emerson, which would of course eventually become Emerson, Lake and Palmer, at the time Mitch Mitchell and Jimi Hendrix of the Experience were scheduled to audition with the group. Tragically Hendrix ODed two weeks before the audition. I still wonder at what that supergroup might have produced