In J. Tate's "Relations between $K_2$ and Galois Cohomology" Lemma 3.4 J. Tate talks about the transfer map in k-theory, and references the Milnor's "Introduction to Algebraic K-Theory", but in the proof of the mentioned lemma Tate uses a property of $K_2$-transfer $$f(\text{tr}(y))=\sum_{s\in G}sy$$ In the milnor's book there's not a proof of this fact, moreover, there's not a mention for the action of $G=\text{Gal}(L/F)$. Then, how acts G over $K_2$?, for the Matsumoto's Theorem, $K_2L$ has a presentation by the symbols $\{\alpha,\beta\}$ with $\alpha,\beta\in L^{\bullet}$. How acts $G$ over the symbols?.
Any hint for the proof of the formula above?
Thanks.