This might relate to Elizabeth (born 1871),

From Glos BMD

Child Surname Child Forename Father Surname Mother Surname Mother's Former Name Year District Office Register Entry
ADAMS Elizabeth [none] ADAMS CUTTER 1871 Forest of Dean Monmouth, Coleford 27 272

And then from this site,

Record_ID 151831
Entry_Number 419
Year 1873
Month Mar
Day 6
Parents_Surname ADAMS
Child_Forenames Elizabeth
Fathers_Forenames [not stated]
Mothers_Forenames Elizabeth
Mothers_Surname
Residence Shortstanding
Occupation [not stated]
Officiating_Minister W.H. Taylor
Event Baptism
Memoranda
Notes
Register_Reference PFC82 IN 1/4
Page_Number 53
Parish_Chapel Christchurch

No father’s name is given in either case. If William ADAMS had been the father would this not have been stated?