  Let G be a split real Kac-Moody group of arbitrary type and let K be its
maximal compact subgroup, i.e. the subgroup of elements fixed by a
Cartan-Chevalley involution of G. We construct non-trivial spin covers of K,
thus confirming a conjecture by Damour and Hillmann (arXiv:0906.3116). For
irreducible simply laced diagrams and for all spherical diagrams these spin
covers are two-fold central extensions of K. For more complicated irreducible
diagrams these spin covers are central extensions by a finite 2-group of
possibly larger cardinality. Our construction is amalgam-theoretic and makes
use of the generalized spin representations of maximal compact subalgebras of
split real Kac-Moody algebras studied in arXiv:1110.5576. Our spin covers
contain what we call spin-extended Weyl groups which admit a presentation by
generators and relations obtained from the one for extended Weyl groups by
relaxing the condition on the generators so that only their eighth powers are
required to be trivial.
