Students often complain that they don't like group work because they wind up having to do "all of the work themselves."1 They obviously haven't figured out the game.
Once assigned to a group, your group mates will immediately begin to smell for signs of weakness: the weakness of being willing to do more than your fair share of the work. There are some pretty obvious signals of this, like being the first one to organize a meet-up time or saying that you "really want to do well on this assignment."
Ostensibly the other group members will agree with this, and in fact2 they may even convince themselves that they really want to work hard too. At first. But when it comes to crunch time, why would they bother doing the work when you have clearly signaled that you would ultimately be willing to do it for them?
The key to avoid doing excess grunt work is to counter-signal from the get-go. Don't fool yourself into thinking that you can pressure others to do more work, as you hold none of the carrots. The more earnest you are, the more moral hazard you give to your fellow group mates to do less work. Even if you do want to work hard and do well, don't make a big show of it and give others an incentive to slack off.
Teachers could avoid this problem easily by having students grade one another on their effort during the group assignment. But this is far too unconventional and most teachers wisely worry more about their general perception by students than actual fairness3. So the status quo remains.
1: I.e., the phrase "I hate group work" has 115k Google hits.
2: Their far mode mind system is active here, so they're thinking about their morals and what kind of person they generally want to be, instead of the details that would go into actually doing the work.
3: Teachers do this to minimize the probability of low student ratings at the end of the year, as they are generally risk averse and want to keep their jobs.