Gerry,
Yes, I have ancestors Cornelius and Hanora Hayes who arrived as Assisted Immigrants to Vic in Apr 1854 per 'Merchantman' from Tipperary. Within a couple of years they were in Maitland, NSW where Honora's brother, Roger Kennedy, was a police man. They remained at Maitland, then Singleton, NSW until their deaths: Hanora in 1868 and Cornelius in 1871. Their four children were Mary Ann, Bridget, Honora and Joseph. I don't think there is any Elizabeth in the family. On the 'Merchantman', also as Assist Immigrants from Tipperary, were Patrick and Margaret Hayes & children. I have not proven a connection with my Cornelius and Hanora, although possible. I might add that I have not researched Patrick and Margaret though. So unless your Elizabeth is connected to Patrick and Margaret I don't think that she would be connected to my Cornelius and Hanora.
Good luck with your research and thank you for taking an interest in my query.
Anniebell