I'm not the most well read in terms of philosophy so apologies if this question comes off as very beginner.
I've been thinking about the 'murderer at the door' thought experiment from the viewpoint of Kant's deontology and how it would apply if the person given the choice to lie were instead an artificial intelligence, specifically an artificial narrow intelligence (one that only performs well in one specific domain). The person who the murderer wishes to kill is still human in this argument.
A Kantian might argue that under the first formulation of the categorical imperative that and AI shouldn’t lie because in a world where all moral AI’s lie, AI would be of no use to us and thus we wouldn't create them. However if an AI is programmed to make moral decisions and nothing else under Kant's second formulation, it cannot have any ends and thus it's actions have no moral value either way. Finally Kant's final formulation is based on autonomy, but an autonomy implies that the moral agent is able to make a different actions put in the same scenario many times. However at a fundamental level a machine is deterministic so in fact it should respect the human's autonomy above it's own and lie to the murderer.
So my question is this, if each formulation works so differently in such a dilemma does that mean that this moral law simply cannot apply to AI in the same way or is there some other way to resolve these discrepancies. If it's not possible to resolve this then does this mean that Kant's claim that rationality is the basis on which morality can be built does not apply to AI no matter how rational they may be. Or is it simply that an AI cannot be truly moral if it is a narrow AI.