Let Fn be the free group on n generators, and PΣn be the group of automorphisms of Fn which send each generator to a conjugate of itself. Let Kn be the kernel of the homomorphism PΣn → PΣn−1 induced by mapping one of the free group generators to the identity. We show that Kn has cohomological dimension (n − 1), and that Hi(Kn, Z) is infinitely generated for 2 ≤ i ≤ (n − 1).