Why? "they" refers to the group members, because "they" is a plural pronoun.
"It changed its mind" would be how you could write the first phrase. In this phrase, "it", a singular pronoun, would refer to the group.
They all put on their jackets.The main ambiquity in the phrase is that 'their' isn't -technically- correct as a gender neutral third person possessive pronoun, despite its common usage in that form. (Say -that- five times fast!) If you replace 'their' with 'his or her', the agreement makes much more sense:
Everybody changed his or her mind.
They all changed their minds.
"The group changed its mind"
and
"The group changed their minds"
depending on the meaning you want the statement to have. Are you emphasizing the concept of "group-think" or the independent conclusions of the individuals?
It's more poetic than
"The members of the group changed their minds".
or
"Each group member changed his or her mind".
But at least you get a ridiculous bag of options for a single sentence.