I wonder if Charles's surname was simply missed off the 1881 census??
FreeBMD has the following marriage including
Charles Frederick Adams and Caroline Wilson
Sep Qtr 1876 Birmingham 6d 307
There is this Charles Fredk Adams and family in 1861 in Chester??
Sarah Adams Married 56 b Essex
Martha Jane Adams 18 Milliner b Cambridge
Samuel Law Adams 16 Railway Clerk b Cambridge
Charles Fredk Adams 12 Chester
Maria Pennington 15 Servant
And 1851 ?
https://familysearch.org/ark:/61903/1:1:SGVV-V1JKay