No.

Consider an irreducible polynomial $f$ of degree $\geq 6$ with all the nonvanishing coefficients in even degrees. For such a polynomial, for every root $\alpha$, $-\alpha$ is also a root, and thus a Galois conjugate. If $S$ consists of the embeddings that send $\lambda$ to any number of roots together with their negations, then the sum is $0$.

Thus, $2^{ \deg f/2}$ different subsets all lead to the same sum $0$. Some of these are cosets. However, since $\deg f \geq 6$, the subsets of size $\deg f-2$ cannot be cosets as their order doesn't divide $\deg f$, so the map is not injective away from cosets.