In 1901, the family is still at 14 Gertrude Road, Walton on the Hill, but William senior is probably away at sea. It looks as though William Alfred jr was an only child.
1901 Lancashire Walton on the Hill District 28 page 14
Mary L Kermode Wife 37 no occupation given Born Liverpool
William A Kermode Son 18 Architect's pupil Born Liverpool
Does this seem to be the right family ?
JC