Other instances of subgroup rigidity without strong rigidity

Identify other locally compact groups, beyond the groups G_n considered in this paper, in which lattices exhibit subgroup rigidity (any two isomorphic lattices are conjugate by an ambient automorphism) but fail strong rigidity (some lattice automorphisms do not extend to ambient automorphisms).

Background

The paper proves that for G_n = Isom^+(X_n) the lattices display a phenomenon where isomorphic lattices are conjugate under Aut(G_n) (subgroup rigidity), yet some automorphisms of a lattice do not extend to automorphisms of G_n (failure of strong rigidity).

The authors emphasize that this phenomenon does not appear in real or p-adic Lie groups and state they are not aware of other examples, suggesting an open direction to find additional instances or to characterize when such behavior occurs.