It is related to the Unlawful Oaths Act of 1797 (UK). It was a law passed by the British Parlament as a way to tackle the increasing number of mutiny incidents (See: Nore Mutiny and Spithead Mutiny). The law considered unlawful oath the ones that engaged in any mutinous or sedition purposes. The maximum punishment for this crime was seven years' transportation (transportation was a punishment that consisted to be transferred to one of the U.K. penal colonies to serve the entirely of your sentence - in this case Australia).

So, there is a high possibility that this man was part of a rebel group or he was just a sailor or a soldier that was planning a mutiny.

Source: A history of the criminal law of England by James Fitzjames Stephen. Page 294 Link