Open-cover-to-space transfer for Homp(X) being a topological group

Determine whether Homp(X), the group of homeomorphisms of X with the topology of point-wise convergence, is a topological group under the assumption that X has an open cover U such that Homp(U) is a topological group for every U in the cover.

Background

The author seeks criteria under which the property that Homp(X) is a topological group can be deduced from local data. This question asks whether verifying the property on each member of an open cover suffices to guarantee it for the entire space.

Such a result would provide a powerful local-to-global principle for the structure of homeomorphism groups with the pointwise convergence topology.