Say a topological group $G$ is *topologically characteristically simple* if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here `automorphisms' are required to preserve the topology as well as the group structure).

Q1: Has someone written down a classification of locally compact second-countable abelian groups that are topologically characteristically simple?

I tried to derive a classification myself; the case I am having difficulty with is when $G$ is torsion-free and the group $pG$ of $p$-th powers in $G$ is a proper dense subgroup of $G$. (Another possible division of this case is between those $G$ with a dense divisible subgroup, and those with no divisible subgroup, but I don't know where to go from there.)

I know one such group for each $p$: let $G$ be the group of all functions from $\mathbb{N}$ to $\mathbb{Q}_p$ under pointwise addition such that all but finitely many values are in $\mathbb{Z}_p$, and topologise it so that the group of functions from $\mathbb{N}$ to $\mathbb{Z}_p$ is an open subgroup with the compact-open topology. This example is topologically characteristically simple with a dense divisible subgroup (the finitely supported functions).

Q2: Are there any other locally compact second-countable abelian groups $G$ such that $G$ is torsion-free, $pG$ is a proper dense subgroup (edit: and $x^{p^n} \rightarrow 1$ for all $x \in G$)? If so, are any of them topologically characteristically simple?