Definition: Topological compactification is such compactification of a topological space such that all autohomeomorphisms on that space can be continuously extended into autohomeomorphisms of the compactification.
I have seen such compactifications called H-compactifications, Equivariant extenstions or G-compactifications (where G could be a subgroup of the group of all autohomeomorphisms of the studied space).
Which of this terms is the most used in your opinion?