NSW BDM definitely show June as Armstrong for the marriage. She is not listed as MILLS in 1943 at the same address as Sophie (NSW). I found her listed as ARMSTRONG in 1954 (district of Watson, sub district Kensington) as a dressmaker which is the same occupation as her later entry as WELCH. The other Armstrong's at the same address might have been her in-laws but there was not a John.
Electoral rolls show Sophie at Kyneton (Victoria) in 1914 just 2 years prior to her marriage. Then there is a gap all the way til 1943 in NSW so maybe June's birth and marriage (to Armstrong) might have been in SA which is where Sophie was born in 1890 although Sophie's parents, Charles UNMACK (a butcher) and Lillie/Lilly/Lily Elizabeth FRY had their marriage registered in Victoria that same year. The 5 children born after Sophie were all registered in Bendigo.
Sophie is not on electoral rolls after 1963 and there is a possible death registered at Sutherland NSW for 1965 - parents not named and only as Sophie MILLS. In the 1943 and 1954 rolls she is the only MILLS listed at her address. There is a George MILLS listed with her in 1949 and 1958 but unknown if he is the husband or maybe a son. In 1963 she is alone again. George's name is too common to determine his death. You would need to purchase the marriage certificate to have a chance of determining his DOB and parents.
If Charles UNMACK was in fact Charles Julius then his death was registered at Bendigo in 1917. Lily/Lilly was still registered at Bendigo in 1914 then a gap til 1930 when she is found in Sydney with her son, Julius Harold (he died 1967 St Leonards NSW). She is not registered after 1933. Found what could be her death but as Elizabeth J UNMACK in Nth Sydney in 1935. This shows the parents as William and Amelia, so going back to birth indexes there is a birth for an Elizabeth Jane FRY 1868 Kangarilla, SA to William FRY and Amelia PEDLER. So maybe Lily was just a nickname?
Unfortunately none of this actually helps you find June's marriage to John ARMSTRONG but a bit extra for her background doesn't hurt.
Hope this helps rather than hinders!
Sandra