In a previous paper the author determined all possible cohomological invariants of Aut(J)-torsors in Galois cohomology with mod 2 coefficients, where J is a split central simple Jordan algebra of odd degree n⩾3. In the present paper we extend these results to a larger class of groups indexed by r=0, 1, 2 or 3, and ni positive integers that add to n⩾3 such that n1 is odd. We also give precise values for the essential dimension of these groups.