Maps of Ostaszewski and related spaces

Under ♦ there is an Ostaszewski space which is retractive, homeomorphic to every uncountable closed subspace, and homeomorphic to every locally countable regular Hausdorff uncountable continuous image, and also an Ostaszewski space with none of these properties. There is a ZFC example of a thin-tall locally compact scattered space for which each nonempty Cantor-Bendixson remainder is a retract. Attention is also paid to robustness under various types of forcing.