I want to use the transfer maps to calculate the cohomology of the symmetric group $\mathfrak{S}_3$ with integer coefficients. Consider a $2$-Sylow group $H:=\langle(0,1)\rangle$ and the $3$-Sylow group $\mathfrak{A}_3$. We then know that
$$H^k(\mathfrak{S}_3) \cong \overline{H}^k(H)\oplus \overline{H}^k(\mathfrak{A}_3),$$ where $\overline{H}$ is the subgroup of stable classes. Since $H$ and $\mathfrak{A}_3$ are cyclic groups, their cohomology is easy, my problem is to find which classes are stable under the conjugation morphisms. Any ideas how to do this?