Let $K/{\Bbb Q}$ be a finite Galois extension, and ${\mathrm{Gal}}(K/{\Bbb Q})$ the galois group. The ${\mathrm{Gal}}(K/{\Bbb Q})$ module ${\text{H}}^1({\mathrm{Gal}}(\overline{\Bbb Q}/K), {\Bbb Z}/n{\Bbb Z})$ gives the canonical isomorphism \begin{equation*} {\text{H}}^1({\mathrm{Gal}}(\overline{\Bbb Q}/K), {\Bbb Z}/n{\Bbb Z}) \cong {\mathrm{Hom}}({\mathrm{Gal}}(\overline{\Bbb Q}/K), {\Bbb Z}/n{\Bbb Z}). \end{equation*} By duality we obtain the ${\mathrm{Gal}}(K/{\Bbb Q})$-module ${\mathrm{Gal}}(\overline{\Bbb Q}/K)^{\mathrm{ab}} \colon= {\mathrm{Gal}}(\overline{\Bbb Q}/K)/[{\mathrm{Gal}}(\overline{\Bbb Q}/K) \colon {\mathrm{Gal}}(\overline{\Bbb Q}/K)]$.
Q. Which is the correct action of $\sigma \in {\mathrm{Gal}}(K/{\Bbb Q})$ on an element $\tau \in {\mathrm{Gal}}(\overline{\Bbb Q}/K)^{\mathrm{ab}}$, $\tilde{\sigma} \tau \tilde{\sigma}^{-1}$ or $\tilde{\sigma}^{-1} \tau \tilde{\sigma}$, where $\tilde{\sigma}$ is the extension of $\sigma$ to ${\mathrm{Gal}}(\overline{\Bbb Q}/{\Bbb Q})$ ?