The first set of chicks that produced that black pullet was in May.
Then about 3 months ago, Karen went broody and set on some eggs. All of those have white barring. Of course, except for the father, I have no idea who the other parent(s) is or are.
Barring is inherited differently than many other genes, because it is on the Z sex chromosome.
A rooster has chromosomes ZZ, so he can have barring on both of them, or one of them, or none of them. If he has barring on both Z chromosomes, he will give it to all his chicks. I was guessing from the photos that your rooster may have barring on just one Z chromosome, which would mean he could give barring or not-barring to any chick. (2 barring genes makes a rooster with more white: either wider bars or more bars of white.)
A hen has chromosomes ZW, so she can only have one barring gene. She gives that barring gene to her sons, and gives a W chromosome to her daughters (makes them female, does not give them barring.) A hen inherits her W chromosome from her mother, and her Z chromosome from her father (so it only matters whether her father has barring, not her mother.) A hen can never have the lighter (more white) kind of barring, because she can never have two barring genes.
So if the barred rooster has 2 genes for barring, then every chick of his will have barring, and he cannot be the father of the black pullet.
But if the barred rooster has just one gene for barring, then half his chicks will get barring from him, and the other half will not. So this would mean he could produce some black daughters.
Then about 3 months ago, Karen went broody and set on some eggs. All of those have white barring. Of course, except for the father, I have no idea who the other parent(s) is or are.
Male chicks can inherit barring from their father or their mother or both, so it would also be possible to get barred males from barred hens with the beta rooster. (Although I think the beta rooster has a gene that turns black into white, but I expect him to give that to half his chicks rather than all of them. That would let half his chicks show black in their coloring.)
Here is the daughter.
If I get a chance I will take a picture of an open wing to expose the white feather.
She is pretty! Unfortunately, I don't know which parents she came from.
I suppose if you really wanted to try to figure it out, you could pen the males and females in specific combinations for a month or more, hatch eggs from each group (sorted by which hen laid the eggs), then look at the chicks and decide whether to try other combinations.
The two biggest factors I am not certain about: whether the barred rooster has two barring genes (cannot produce un-barred chicks) or one barring gene (can produce both barred and unbarred chicks) and whether the beta rooster has one or two of the gene Dominant White that turns black to white (two of it, he cannot produce chicks that show black, but one of it means he can show some chicks that show black and some that show white instead of black.)
