So she may not be the daughter of Lizzie Hackett, but the daughter of Charles Firman and his first wife.
Not sure of the birth record, but there is a death record in NSW BDM of Elizabeth Emily L Jones, dau of Charles Firman and Elizabeth Emily L Registration no. 9106/1960, Sydney District.
An Elizabeth Emily Louise Jones, is residing at 2 Kelburn Rd, Roseville in the 1954 Electoral Roll, which is the same address as Allan Hackett Jones, medical practitioner, and I can't see any record to say that he married, so this could be his sister (half-sister)?