Is bullying hereditary?

My son is a bit of a bully. He is 6 yrs old and excludes people, tells kids he doesn’t like them, and plays with only 2 kids making sure the other kids know he doesn’t like them. Both my husband and I encourage kindness, we have read books, listened to podcasts, arranged play dates, and we are still struggling with trying to get him to believe that kindness is cool… cause we think it is. Both of us have been bullied in school. My MIL on the other hand is very much a bully. still… he doesn’t spend too much time with her, but I do see them at odds at times as they try to out bully each other… could he have inherited a bit of a “mean” streak from her? Oh to combat this? Also, we can’t do councelling until he is 8, where I am at.. but we fully intend to do so…