Locally finite closed cover 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 a locally finite closed cover C such that Homp(F) is a topological group for every closed set F in the cover.

Background

Extending the local-to-global theme, this question asks whether the property of Homp(X) being a topological group can be inferred from its validity on each member of a locally finite closed cover. Locally finite covers often enable patching arguments in topology, making this a natural setting to investigate.

An affirmative answer would broaden methods for constructing spaces with well-behaved homeomorphism groups by piecing together closed subspaces with controlled automorphism group topologies.