Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}_F$ and $\mathcal{O}_K$ be the associated rings of integers, and let $n\geq 1$.

When is $$ K_{2n-1}(\mathcal{O}_F)^G \cong K_{2n-1}(\mathcal{O}_K)? $$

It would be good enough for me to have this on the prime-to-2 parts. So let $p$ be an odd prime. Then $$ K_{2n-1}(\mathcal{O}_F)\otimes_{\mathbb{Z}}\mathbb{Z}_p\cong H_{ét}^1(\mathcal{O}_F,\mathbb{Z}_p(n)) $$ by the Bloch-Kato conjecture, as proven by Rost, Suslin, Voevodsky and Weibel. So my question becomes:

when is $$ H_{ét}^1(\mathcal{O}_F,\mathbb{Z}_p(n))^G \cong H_{ét}^1(\mathcal{O}_K,\mathbb{Z}_p(n))? $$

A concise reference to a place that summarises everything we know about this would be great! Or maybe it is clear to the experts that this is always true? That would be even better.