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