Let $G$ be a profinite group which fits in the following short exact sequence: $$ 1\rightarrow N\rightarrow G \rightarrow K\rightarrow 1 $$ Assume that $N$ is a pro-$p$ group and that $K$ is topologically finitely generated. Note that $K$ acts naturally by conjugation on $N^{ab}$ and thus we way view $N^{ab}$ as a $\mathbf{Z}_p[[K]]$-module.

Q: If $N^{ab}$ is a finite type $\mathbf{Z}_p[[K]]$-module, does it follow that $G$ is topologically finitely generated?

Let $G$ be a profinite group which fits into the following short exact sequence: $$ 1\rightarrow N\rightarrow G \rightarrow K\rightarrow 1 $$ Assume that $N$ is a pro-$p$ group and that $K$ is topologically finitely generated. Note that $K$ acts naturally by conjugation on $N^{ab}$ and thus we way view $N^{ab}$ as a $\mathbf{Z}_p[[K]]$-module.

Q: If $N^{ab}$ is a finite type $\mathbf{Z}_p[[K]]$-module, does it follow that $G$ is topologically finitely generated?