Abstract In this article we explore some properties of H-structures which are introduced in [2]. We describe a construction of H-structures based on one-dimensional asymptotic classes which preserves pseudofiniteness. That is, the H-structures we construct are ultraproducts of finite structures. We also prove that under the assumption that the base theory is supersimple of SU-rank one, there are no new definable groups in H-structures. This improves the corresponding result in [2].