In 2001 I wrote to the Comune di Santa Sofia using an online translation site and sent money & received the marriage certificate of my grand-parents and confirmation of my grand-father's medical degree and the birth certificate of my mother. But I have not been able to go any further back . I have also found the online translation of the letter and documents very difficult to understand however have interpreted the following:
Carlo Pio Ferdinando Baldassarre Gregorio Franceschi born 12 March 1865.
parents Cosimo Franceschi also a doctor / surgeon in Florence. (Both were directors of hospitals at some stage.) and mother Amalia Malenchini.
Carlo married Maria Leoni on 13 December 1897 in Santa Sofia
They moved to Australia in 1900 or 1901 but I cannot find any shipping records. My mother's birth certificate indicates they married three years after her birth which all the family refuse to believe (!) One family member paid a genealogist in Italy and received totally different information to what I have however I believe the information I received to be accurate.
Any suggestions where I can find further family history?
I was not able to obtain any further birth records.

Perhaps this is a record you need?

Title
Carlo Franceschi - Naturalization
Contents date range
1907 - 1915
Series number
A1
Click to see which government agency or person created this item.
Control symbol
1915/20368
Item barcode
33837
Location
Canberra
Access status
Open
Date of decision
14 Jan 1982
Physical format
PAPER FILES AND DOCUMENTS (allocated at series level)
Date registered
12 Oct 1998

If so there is a 12 page digital document of it on the Australia Archive site indicating he arrived in Sydney on 28th of Dec 1900 aboard the Grosser K. He was born in Florence on the 12 Mar 1865 and is 44 years old living with his wife and four children.

Thank you so much for your response.
I'm sorry, I did not realize there had been any answer to my query.
The naturalization papers are indeed a big find after all this time.
I really appreciate your help.

Unfortunately they don't provide any further information about their ancestry but are, at least, something...

I believe you can order these records from the Australian Archives for a small fee. The records themselves should hold more clues for you.