I've heard a few times that the symmetric group is an algebraic group over a field with one element, and that the alternating group is quite specifically $SO_n(\mathbb{F}_1)$. This does make a lot of sense intuitively, and actually helps to explain to non-specialists the relations between different things I have done.

However, objects over the (non-existent) field with one element aren't just a metaphor - they are objects that can be defined properly, though that may not have happened yet. What is the status for the correspondence between $Alt(n)$ and $SO_n(\mathbb{F}_1)$? Is there really a well-defined homomorphism of some sort, and, if so, are there references where this is worked out?