Female bullies are a function of patriarchy too, because it's a social system based in a dominance hierarchy -not just a gender dynamic. You climb in the pecking order by putting others "in their place" below you. The only way to “win” is if someone else “loses.” This happens with gender power differentials, but also with race, sexuality, and just plain garden variety bullying. And yes, women absolutely do this too, as well participate in upholding patriarchy in various other ways, including, but not limited to:
Maintaining gender binaries; enforcing what is considered appropriate for boys or girls, rather than letting each child be who they naturally are.
Obsession with female appearance and body image; both their own and that of other women.
Shaming men and boys who show emotion or vulnerability.
That's why interrupting this social system would benefit us all, men included.