Let $G$ be a group whose centre $Z(G)$ contains the element $g$. Show that ${\displaystyle {\hat {g}}}$, the action to the permutation group is the identity permutation.
I get that by definition ${\displaystyle {\hat {g}}} = {\displaystyle {\hat {k}}} {\displaystyle {\hat {g}}}{\displaystyle {\hat {k}}}^{-1}$ for $k$ in $G$ but I do not know how to move on.
I use the notation $\sigma$, i.e., $\sigma=\phi(g)=\hat{g}$, which you have defined. Then we have $$ \sigma(k)=gkg^{-1}=kgg^{-1}=k, $$ because $g$ is in the center of $G$. Hence $\sigma(k)$ is the identity.