The first element in the stable homotopy groups of a $K(\mathbb{Z}/2, n)$ (which is outside the range of the Freudenthal suspension theorem) is $\pi_{2n} K(\mathbb{Z}/2, n) \simeq \mathbb{Z}/2$. In particular, there is a unique nontrivial stable map $S^{2n} \to K(\mathbb{Z}/2, n)$. This map is necessarily zero in cohomology.

In what Adams filtration can we detect this map? (Can we detect it using some other cohomology theory?)