I am interested in examples of the following property, where $G$ is a non-discrete locally compact topological group:

(*) The open normal subgroups of $G$ have trivial intersection, but $G$ has an open normal subgroup $H$ such that for all open normal subgroups $K$ of $G$, there is an automorphism $\alpha$ of $G$ such that $\alpha(H) < K$.

Notice that $G/H$ is then non-Hopfian in an interesting way: for all open normal subgroups $K$ of $G$ such that $K \le H$, there are non-injective epimorphisms between $G/H$ and $G/K$ in both directions.

For instance, $G = \mathbb{Q}_p$ and $H = \mathbb{Z}_p$ have this property, and the quotient is one of the standard examples of a non-Hopfian group, namely the Prüfer $p$-group. I am sure there are examples as well where $G$ is some sort of product of infinitely many copies of a group and there are automorphisms acting by 'shifts'. But I am wondering if there are more exotic constructions, where for instance all the open subgroups have trivial centralisers, or none of the open normal subgroups are compact.

(My specific interest is to find an example where $G$ has a hereditarily just infinite open subgroup that is not virtually abelian, or to show that this cannot happen, relating to the discussion of locally h.j.i. groups in section 5 of this paper of Barnea, Ershov and Weigel: http://arxiv.org/abs/0810.2060)