I have a bully child. His biggest victim is another of my children, which is somewhat a horrifying situation to be in as parent. I don't have the words to describe how that makes me feel.
He is NOT bullied at home, we are strict as his behaviour is intolerable but he is also one of four sons and none of the others are like it. We have however in the past been threatened by his victims, to the extent that several years ago I was too scared to do school run alone, or even answer the door to unexpected visitors.
I think I know a little about parenting, I worked for a parenting charity for a while hands on, and Social Services repeatedly assess me as a good parent and not in need of support when I beg for help.
However, several years down the line I KNOW what is going on. It's a mix. Firstly, ds1 has Asperger's and it went unsupported by school for far too long, leaving him vulnerable to bullying himself which became a source of fun amongst his classmates. Unlike many children with ASD who have plenty of emotional empathy and struggle to show it, ds1 happily admits to having none, combining it with plenty of cognitive empathy, which means he has the power to manipulate and indeed has seen off a few TAs at the specialist ASD school he now attends.
I could have made a start on helping him years earlier (I have almost completed an MA in Autism primarily as a means of helping him and his ASD siblings) if I'd been backed by school, but I was lied to and made to feel an idiot. Via the MA I met teachers at schools that can actually help him, he could have had that input years before.
I have no idea how to sort the home situation except to evict him if he is doing the same in 4 years when he turns 18, as his sibling is more severe and vulnerable, but at school with help we are getting there. I am sure there are plenty of people whose kids attended school with him who think I must be a bully parent though, and blank me or worse- even though i fought endlessly for help and supervision for him.