Goodman in Kent, Sussex, or London
This is an interesting story.
My great-grandmother was fostered, which means we knew very little about her real family, in fact, nearly nothing. After a bit of lateral thinking, and tracing what we did know, her adoptive families name, and the town (Tonbridge), we found her with her adoptive family. This listed her with her birth surname - Goodman.
She was born in 1909, in Northfleet, with her birth registered in Bromley. There were no formal adoptions or legal paperwork involved at this time. She never looked for her birth family - when she was about 2, the lady who came to visit her at her foster family home (well dressed, would arrive in a horse and carriage and give her new parents money.) said she would no longer be coming to see her, because she was beginning to form an attachment and recognise her. This could have been her mother or maybe more likely her maternal grandmother. My great grandmothers idea was that she was the result of an upstairs/downstairs relationship. It's fairly certain she was illegitimate.
This means that there may be out there, someone with a Goodman tree who has children they cannot find (Children born alive, children died which appears on some census records). There were Goodman's in the Northfleet/Gravesend area around the time my great grandmother was born, who could have been relations of her mother. I know that Goodman was her mothers name because illegitimate children had their mothers surname, even if they knew who the father was.