Here’s the baptism details
Frank William Edward Rogers
Birth date 12 Oct 1911
Baptism date 05 Jul 1912
Denomination Anglican
Place Nuneaton, St Mary
Father Edward Rogers
Mother Kate
Residence 50 Abbey St
Father's occupation Hairdresser
Frank died 1987 - you can search for them on 1939 Register
Census says 1 child born 1 died but I can’t find a birth before Frank

Wonder why they never registered her as Rogers.
Probably because they were not married - will be interesting to see who registered the birth
Edit to add - found this
common occurrence was the illegitimate child. It was implied, though not always the case, that the omission of the father’s name from the certificate suggested illegitimacy.
From 1875 the reputed father HAD to be present at the registration to consent to his name being added.