Annie Rye seems to have been in Lancashire for a fairly short time:
birth of George E Rye, mmn Cottingham registered 1920 Q2 Elham Kent
burial of George Edward Rye, January 15th 1921 Parr, St Helens, Lancashire
birth of Sidney G Rye, mmn Cottingham registered 1921 Q4 Elham Kent
birth of Sidney G Rye, mmn Cottingham registered 1921 Q3 Prescot, Lancashire
added: I messed this up, not sure how. The next child was born in Elham however.