Opponents of EMT have repeatedly argued that the functional roles that define a mental state must be understood in a fine-grained way ...Actually, this coarse-grained/fine-grained stuff is an innovation from the EC side of things. It's a distinction that ECists have introduced (but left in intuitive terms) to fend off the observations (which seem to be universally accepted*) that there are differences between Inga and Otto that brain-o-centric cognitive psychologists have cared about. (Indeed, Inga and Otto don't even behave the same way. Otto will "remember" much more reliably than will Inga. As the story is typically told, his memories do not fade over time.)
That aside, my sense is that some history of these arguments has been lost. Ten+ years ago, one often encountered the claim that there is no principled reason for thinking that cognitive processes take place in the brain. But, if one can understand cognitive processes as cognitive psychologists do (as what Clark, Kiverstein, Farina, et al., describe as"fine-grained"), then there is some principled reason to think that cognitive processes take place in the brain. So, A&A don't say that functional roles, or the individuation of cognitive states, must be fine-grained. Only that the states brain-o-centric cognitive psychologists study are such-and-so (which Clark, Kiverstein, Farina, et al.will describe as fine-grained). That's why they are principled, brain-o-centric cognitive psychologists.
Or, think of it this way. We can draw a distinction between an hypothesis of fine-grained cognition and a hypothesis of coarse-grained cognition. The first, by all accounts, would seem not to be extended, where the latter might be. But, that's fine by me. The old-fashioned cognitive psychology that studies brain processes could go on as before.
* Later in their text, K&F write, "In responding to opponents of EMT, we conceded above that there are undeniable (fine-grained) functional differences between the processes that take place inside a person’s head and extended processes. " Excellent.