Putting 'Balberman' in the fuzzy category of name searches, I came up with many results. However, I concluded that Golda Balbermann was the closest because 1) she was born in 1929, the right age to have signed the book, and 2) because the name was only one letter off, and the closest match.
There was also a Gesia Balbermann (b. 1924) that came up from the Czestochowa Forced Laborers database, so we cannot be sure, however I think it was Golda that signed the book.
Blima (b. 1891) was, I think, Golda's mother (but could be her aunt as well), and her siblings were Perla (b. 1916, although given the age difference, perhaps a cousin?), Rywka (b. 1921), Berta (b. 1927), and Dworja (b. 1932).
They all have the same deportation date, March 17, 1942, which was during the month of daily deportations to Chelmno.