Original: The Council of the Isles of Scilly is a separate unitary authority to the Cornwall Council unitary authority, and as such the islands are not part of the administrative county of Cornwall.
Simplified: The islands have had a unitary authority council since 1890, and are separate from the Cornwall unitary authority, but some services are combined with Cornwall and the islands are still part of the ceremonial county of Cornwall. This council is part of the UK and currently known as the Council of the Isles of Scilly.