No official records exist for Irish people leaving Ireland prior to 1922 as we were part of the United Kingdom.
Most Irish people emigrating to Australia in 1901/1911 would have gone to England first via domestic ships before making the big jump to Australia.
Lilly's surname is central to finding her family & information.
I have no information on Lily Lernan or Lilly Lerhinan emigrating. However records exist for Rowland Lernan/Lerhinan born in County Longford circa 1887 who gave Clonawhite,Co Clare as his home address when he joined the Irish Guards Regiment in Liverpool and died in the first world war in 1918. You might have some luck checking official records of passengers on board ship in Britain called information sheet 31 from the Maritimes Archives & Library. Search the internet for information sheet 31 passengers.