You can see the other objections here and here.
I have dubbed the following objection the epiphenomenalism objection. Although it is logically possible for the mind to be epiphenomenal (to have no effect on actual behavior) many find that conclusion to be unacceptable and to warrant discarding any theory that would lead to it. Once again we rely on the premise that externalism involves a non-causal connection between the mind and the external world (for reasons discussed earlier, as well as another reason I will present below). If the mind as described by externalism really isn’t epiphenomenal then it must be a cause of behavior, and so if we work backwards from behavior we should encounter something that can be considered the externalist mind (however an externalist chooses to define it). Immediately of course behavior is caused by the activity of neurons, which in turn are caused by the activity of other neurons. No matter how far back we follow these causes we never have evidence of a non-causal connection, only purely causal connections between neurons and other neurons, and neurons and sensory inputs. This in turn implies that either the non-causal connections that are part of externalism are not part of the mind, which really is the cause of behavior (and this is effectively a denial of externalism), or that the non-causal connections are part of the mind, but that this mind isn’t a cause of behavior (epiphenomenalism). Both these conclusions seem to imply that externalism should be dropped in favor of some other theory.
I should mention that there is also a “split-mind” possibility, namely that some of the mind is responsible for behavior, and is found by our investigation into behavior’s causes, and that there is another part of the mind that the non-causal connections are part of, but that it is not connected to the part of the mind that controls behavior. I reject this possibility because it still leads to the conclusion that the mind is at least partly epiphenomenal, and more strangely that it is composed of two completely separate parts that don’t affect each other (this is not the same as the right-left hemisphere division because the two hemispheres do communicate, through both the corpus callosum and the hindbrain).
At this point you may again be considering the possibility that connection between the mind and the external world in externalism is causal. Although I already presented an argument against that possibility allow me to give another here. Let us then consider a causal version of externalism a real possibility. We then say that the mind is connected to an object, say X, by X’s causal influence on the mind. Let’s examine then what this connection means at a given instant of time. Certainly when we look back to the moment of time where whatever information about X that influenced that mind was generated it might seem reasonable to say that our future thought about X is dependant on that X in the past and the mind in the past. However as we consider moments in time closer to the actual thought X falls out of the picture, and instead what we consider to be connected to us is the medium by which the information about X was transmitted (say the reflected light). When we reach the moment of the actual thought there is nothing left from X that we can consider to have a causal effect on us, because the effect of X on us happened some time in the past. When considering that moment of time, the moment when the thought happens, all we can say is that we have a thought about X, even though the thought isn’t connected to X. For all intents and purposes this is the same as internalism.