Following the discussion I have with Yves Cornulier in the following question Finiteness theorems for profinite groups, I would like to ask the following: Suppose $K$ and $N$ are two profinite groups and $K$ acts on $N$. Suppose further that each element of $K$ acts continuously on $N$. We can form the semidirect product $G=N \rtimes K$. If $N$ is characteristc based, that is $N$ has a base for its topology at the identity made of open characteristic subgroups, then $G$ has a structure of a profinite group such that the induce topology on $N$ and $K$ as subgroups is their original topology. In particular, this is the case if $N$ is finitely generated.

Could you give an example where $G$ does not have such a structure? More specifically, could you give an example where $K$ and $N$ are pro-$p$ groups, but $[N,K]=N$?