  We study a symplectic variant of algebraic $K$-theory of the integers, which
comes equipped with a canonical action of the absolute Galois group of
$\mathbf{Q}$. We compute this action explicitly. The representations we see are
extensions of Tate twists $\mathbf{Z}_p(2k-1)$ by a trivial representation, and
we characterize them by a universal property among such extensions.
  The key tool in the proof is the theory of complex multiplication for abelian
varieties.
