I note that there are two Fred’s born Rotherham about 1887 on each of the census returns you mention, I’m not clear exactly what you require but is this him below?
1911 The Green Whiston Rotherham
Fred Rodgers 24 Elsie Rodgers 21 Fred Rodgers 1 Marion Rodgers 0
1901 – 17 Carlin Avenue, Earl Ward, Rotherham All Saints.
1901 Fred Rodgers
Mother Mary A Rodgers
Father Fred Rodgers
Sibling Thomas and others
Birth abt 1887 Rotherham, Yorkshire,
1891 family as above 5 Eastwood Mount, Earl Ward,.Rotherham All Saints.
Colin