As a data point, we missed a connection earlier in the month, and the system automagically rebooked us.
On our original flight, we were both group 4, presumably because I bought the tickets with my Citi AA Executive card.
On the rebooked flight, I was 4 and my companion was 6. I'm guessing we lost the credit card link somehow, and the new groups were status-based (I'm gold and she isn't)?