The full braid group on $n$ strands $B_n$ admits a surjective homomorphism $p\colon\thinspace B_n\to \Sigma_n$ onto the symmetric group on $n$ letters, which takes a braid to the induced permutation of its ends. The kernel $P_n$ is well understood; it is the pure braid group on $n$ strands.

What about $p^{-1}(A_n)$, where $A_n$ is the alternating group on $n$ letters? Let me call this group $E_n$ for now, because I think it should be called the even braid group. However an internet search using this name (and others such as "orientation preserving braids", "positive braids" and so on) came up blank.

Does this group $E_n$ have a name, and has it been studied anywhere in the literature?

I would be particularly interested in computations of the cohomology rings of these groups.