Sorry- I'm confused.
1851 HO107; Piece: 2056; Folio: 285; Page: 7
?Court Church Street (feint writing)
Henry Hunter 58 yrs Porter
Elizabeth Hunter 48 yrs
Eliza Hunter 14 yrs
Henry Hunter 11 yrs
Mary Hunter 7 yrs
Sarah Hunter 5 yrs
Josiah Hunter 1 yr
1851 1851 HO107; Piece: 2061; Folio: 198; Page: 21
198 (?) Heneage (?) Street Aston
Elizabeth Hunter widow 39 yrs Dressmaker
Mary Ann 15 yrs
Alice 13 yrs
Elizabeth 4 yrs
Celena 1 yr
1851 HO107; Piece: 2053; Folio: 73; Page: 31
Cordelia Hunter 6yrs visitor to Sara Wrighton
(I thought I'd just look on Family Search- first tried Sarah Hunter - marriage to a Daniel Wrighton 1823 St Martin Birmingham

wonder if that is the connection)
1861 RG9; Piece: 2161; Folio: 5; Page: 4
17 New John Street West Birmingham
Elizabeth Hunter 49 yrs
Henry Hunter 21 yrs Gun Stocker
Cordelia Hunter 16 yrs
Elizabeth Hunter 13 yrs
Selina Hunter 11yrs
So as you say, there are two sets of children both with mothers Elizabeth.
What are the details on Cordelia's birth certificate- how do they compare with Josiah's?