The third homology of $\mathrm{SL}_2(\mathbb{Q})$

We calculate the third homology of $\mathrm{SL}_2(\mathbb{Q})$ with half-integral coefficients. Corresponding to each prime $p$ there is an operator on this group with square the identity. The kernel of the (split surjective) homomorphism to the indecomposable $K_3$ of $\mathbb{Q}$ is a the direct sum over all primes of the $(-1)$-eigenspaces of these operators. The $(-1)$-eigenspace of the operator corresponding to the prime $p$ is cyclic of order the odd part of $p+1$.