Cat,I am also getting confused,you say I have found the right people but I cannot find connection.
Can we clarify,so that I can have another shot at helping you move forward.
You say William Stretton /Rosina Birch
Issue-- Jane 1881 William 1856(do you mean 1886)and Lily 1912
Where did you find these facts?
Lily Strettons mother was called Smith so it could not have been Rosina Birch.
I have found no connection between
Jane 1881
William 1885/6
Lily 1912
Apart from the fact that Rosina was 56 in 1912,it would be most unlikely that she would give birth 31 years after the birth of her first child.( Jane 1881)
If you send for the 1912 birth certificate with the info that Pom found it will go a long way to helping clarify things.