A’Campo showed that the finite symplectic group $Sp(2m,p)$, $p>2$ prime, is a quotient of the braid group $B_n$ for some $m$ depending on $n$. Hence the finite groups $PSp(2m,p)$ are quotients of $B_n$. [These groups are simple][1] non-abelian hence perfect for most $m, p$.   

<cite authors="A’Campo, Norbert">_A’Campo, Norbert_, [**Tresses, monodromie et le groupe symplectique**](https://doi.org/10.1007/BF02566275), Comment. Math. Helv. 54, 318-327 (1979). [ZBL0441.32004](https://zbmath.org/?q=an:0441.32004) [MR0535062](https://mathscinet.ams.org/mathscinet/article?mr=0535062)</cite>


  [1]: https://groupprops.subwiki.org/wiki/Projective_symplectic_group_is_simple