Would not the NZ CAA be the investigating Authority.
Norfolk Island became an an Australian Territory in 1914 and is administered under the Norfolk Island Act 1979, which provides for an Administrator as nominal head of the Norfolk Island Government, a Norfolk Island Legislative Assembly and a Norfolk Island Supreme Court oversighting the Island's legal system (derived from NSW and the UK). The Island's constitutional status is similar to that of the mainland Territories; the major difference is that the Island Government and Legislative Assembly have greater powers and responsibilities, including scope for enacting laws on matters that elsewhere are the preserve of the Federal Government (such as customs, quarantine and immigration).
Whereas Lord Howe island is actually part of NSW.