This is probably something well known (either in the affirmative or in the negative) but I couldn't get this information easily:
Braid group:Symmetric group::?:Signed symmetric group
By "signed symmetric group" I mean the wreath product of the cyclic group of order two by the symmetric group with its usual action as a symmetric group. Equivalently, it is the group of n by n matrices under multiplication where every row and every column has exactly one nonzero entry and that entry could be +1 or -1.
I don't see an obvious way to generalize, nor do I see a reason why no analogue can exist.