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 $H^2_{ \mathcal{ M}}(F, { \bf Z}(i)) \rightarrow H^2_{ \mathcal{ M}}( L, { \bf Z}(i) )^{ G}$.
Categories: math.NT math.KT