Let $p$ be a prime such that the free 2-generator group $B(2,p)$ of exponent $p$ is infinite. Consider the short exact sequence $$ 1\to K \to B(2,p) \to B_0(2,p) \to 1, $$ where $B_0(2,p)$ is the biggest finite $2$-generator group of exponent $p$, which exists by RBP.

**Question.** What is known about the normal structure of the kernel $K$?

More specifically, besides the obvious facts that $K$ is perfect, finitely generated, and of exponent $p$,

Are there any known proper normal subgroups of $K$?

Is $Z(K)=1$?

Could it be that $K$ is in fact simple?