Inheritance of non-topological-group property under h-embeddings into superspaces

Determine whether the failure of Homp(X) to be a topological group is preserved when passing to a superspace Y in which X h-embeds as an open subset. Specifically, assume X h-embeds in Y as an open subset and Homp(X), the group of homeomorphisms of X endowed with the topology of point-wise convergence, is not a topological group. Ascertain whether Homp(Y) must also fail to be a topological group, and analyze the case when X is a dense (or open and dense) subset of Y.

Background

The paper studies when Homp(X), the group of homeomorphisms of a space X with the topology of point-wise convergence, forms a topological group. After identifying broad classes of spaces where Homp(X) is not a topological group (notably those with strong n-homogeneity and no isolated points), the author introduces the notion of h-embedding: a subspace X of Y is h-embedded if every automorphism of X extends to an automorphism of Y.

The question asks whether non-topological-group behavior of Homp(X) necessarily extends to Homp(Y) when X is h-embedded in Y as an open subset, and also in the case where X is dense (or open and dense) in Y. This is motivated by the observation that without a strong condition on the embedding, the property need not transfer, as evidenced by a subsequent positive result constructing superspaces with Homp(Y) a topological group.