Hi Jan,
Found the Murphy family in 1901 at 24 Edinburgh Street, Everton. Noticed difference in ages between the son Frederick and his siblings. Turns out, from looking into the future, that Frederick is a stepson to Thomas.
This is probably his birth:
Births Dec 1893> Davies, Frederick Francis - W. Derby - 8b - 525.
Although, confusingly, there is a Frederick Murphy birth in 1893 as well. But because Emma married Thomas Murphy in 1898, I'd say the Davies birth is the right one and William is also a stepson although he's not with his mother in the future census.
1891:
DAVIES, John - Head - Married - 26 - 1865 - Assistant Storekeeper Iron Foundry - Liverpool.
DAVIES, Emma - Wife - 21 - 1870- Liverpool.
Address: 95 Opie Street, Everton, Liverpool.
RG number: RG12 / Piece:2951 / Folio:64 / Page:10.
That census followed by:
Deaths Sep 1896> Davies, John - 32 - Toxteth Park - 8b - 147.
That census perhaps preceded by a Davies marrying a Davies?
Marriages Sep 1890:
Davies, Emma - Liverpool - 8b - 15.
Davies, John - Liverpool - 8b - 15.
I do think this could be William Davies [AKA Murphy in 1901]:
Regarding his admittance to an industrial school in 1906, he might still be there in 1911. A free search on the National Archives site
http://www.1911census.co.uk/ gave just 2 results as inmates in institutions 'close to home':
William Davies - YoB shewn as 1893 - District West Derby.
Dickiesam