Braid monodromy factorization for a non-prime $K3$ surface branch curve

This paper is the second in a series. The first one describes pillow degenerations of a $K3$ surface with genus $g$. In this paper we study the $(2,2)$-pillow degeneration of a non-prime $K3$ surface and the braid monodromy of the branch curve of the surface with respect to a generic projection onto $\text{C}{\mathrm{\P }}^2$. In future papers we study the fundamental group of the complement of the branch curve and the fundamental group of the Galois cover of the surface with respect to this generic projection.