Can't find him in the 1911 census but hopefully this helps:
GRO index (no link possible)
PERKINS, WILLIAM CHARLES GEORGE HOPKINS
GRO Reference: 1885 D Quarter in LAMBETH Volume 01D Page 503
Corresponding FreeBMD entry:
By 1901 they'd moved to Yorkshire. You can see William's parents were George and Rhoda J and his age of 15 confirms the birth record from 1885.
"England and Wales Census, 1901," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:XS9Y-52W : 29 August 2015), William C G Perkins in household of John W Hogarth, Bridlington, Yorkshire (East Riding), England; from "1901 England, Scotland and Wales census," database and images, findmypast (http://www.findmypast.com : n.d.); citing Bridlington subdistrict, PRO RG 13, The National Archives, Kew, Surrey
Oh you already had that census entry ... sorry. His place of birth is given as Herne Hill which would have fallen under Lambeth's registration district.
Here's his mother's birth record (her mother's maiden name was WHITNALL according to the GRO index)