  Let $L/F$ be a finite Galois extension of number fields with an arbitrary
Galois group $G$. We give an explicit description of the kernel of the natural
map on motivic tame kernels $H^2_{\mathcal{M}}(o_L, {\bf Z}(i))_{G}
{\rightarrow} H^2_{\mathcal{M}}(o_F, {\bf Z}(i))$. Using the link between
motivic cohomology and $K$-theory, we deduce genus formulae for all even
$K$-groups $K_{2i-2}(o_F)$ of the ring of integers. As a by-product, we also
obtain lower bounds for the order of the kernel and cokernel of the functorial
map $H^2_{\mathcal{M}}(F, {\bf Z}(i)) \rightarrow H^2_{\mathcal{M}}( L, {\bf
Z}(i) )^{G}$.
