I’m having the exact same issue. Change avatar works perfectly, as does just viewing the profile. However if I click edit, then I get redirected to a non profile page (in my case I have a page “editorial” that I get directed to instead of the group/1, but I had that behavior at one point when I re-wrote all the rules to use “change-profile” instead of “edit” as the action.)
I have found that if I uncheck the Extended Profiles component from the BuddyPress setup, then it works just fine. So it’s definitely related to xprofiles.
I’m running 3.8.2 and 1.9.2.
Very interested to see if there is a fix to this one. For now I’ve disabled the xprofile component so the users of this site can update their profiles at least.