Considering the profinite integers as the ring of endomorphisms of $\mathbb{Q}/\mathbb{Z}$, I am looking for examples where the kernel of a profinite integer is an infinite subgroup of $\mathbb{Q}/\mathbb{Z}$, preferably non-finitely generated (and how to construct it).
For an ordinary integer $N$ the kernel is simply $\mathbb{Z}/N\mathbb{Z}\subset\mathbb{Q}/\mathbb{Z}$, but I suspect that p-adic integers might do the job (I suspect that any Prufer k-group will be in their kernel for k and p different primes due to the way $\mathbb{Q}/\mathbb{Z}$ and $\hat{\mathbb{Z}}$ can be written in terms of these groups, but I’m not sure about it).