No.1 mkIII cypher marking.

[Bindi2]

What makes you think it is a Lithgow. After WW1 the Brits returned/replaced the rifles that Australia had sent over at the very beginning when supplies were short. Also the Aussies had British rifles issued over there which came home with them. These went through various Lithgow FTR programmes over time till they became surplus and sold.

A photo or more will help.