I can't help but wonder whether we are dealing with two different families here. On the 1901 England census William and Kate A. Kendall are in Leeds, Yorkshire as follows:
William Kendall, head, 44, coal pit sinker, born Newland, Cornwall
Kate A Kendall, wife, 33, born Oldham, Lancashire
Anthony F Kendall, son, 13, born USA Inglewood
Mary B Kendall, daughter, 9, born Hoyland, Bernsley, Yorkshire
William J Kendall, son, 6, born Oldham, Lancashire
Joseph Kendall, son, 4, born Farnsworth, Lancashire
Frederick Kendall, son, 3, born E. Brierly, Bradford, Yorkshire
This information just doesn't match up with the 1911 England census entry for William and Caroline. That entry says they had been married for 18 years which would be about 1893 and it says that Caroline had 4 children (Joe, Ethel, Fred and Sidney) and all were alive.
Jacquie