Closeness of measures on a cardinal

Question

Given an uncountable $\kappa$ and a $\kappa$-complete nontrivial non-normal ultrafilter on $\kappa$, and some $g:\kappa\to\kappa$ with $<_{U}$-rank $\kappa$ (where $f_0<_Uf_1$ iff $\{i<\kappa\, |\, f_0(i)<f_1(i)\}\in U$), we can define a normal ultrafilter $D = \{ X\subseteq \kappa\, |\, g^{-1}(X)\in U\}$ and an embedding $k:Ult(V,D)\to Ult(V,U)$ by $k([f]_D)= [f\circ g]_U$. I see that $\kappa < crit(k) \leq [id_{\kappa}]_U<(2^{\kappa})^+$ and I was wondering if anything more can be said (in ZFC) about $crit(k)$, and otherwise what values of $crit(k)$ are consistent, and for example if it's consistent that $\kappa$ is measurable, there exist non-normal measures on $\kappa$ and always this $crit(k)=[id_{\kappa}]_U$. Basically, I would be glad to be pointed at any result that could make this more clear for me.