When an incident occurs with a G-reg the AAIB investigates with the assistanceof the local authorities no?
If you're talking about an Annex 13 investigation, no.
For an accident or serious incident falling within the remit, Annex 13 says:
5.1 The State of Occurrence shall institute an investigation into the circumstances of the accident and be responsible for the conduct of the investigation, but it may delegate the whole or any part of the conducting of such investigation to another State or a regional accident investigation organization by mutual arrangement and consent. In any event, the State of Occurrence shall use every means to facilitate the investigation.
and
5.18 The State of Registry, the State of the Operator, the State of Design and the State of Manufacture shall each be entitled to appoint an accredited representative to participate in the investigation.