The cohomology of symmetric groups and the Quillen map at odd primes

Gunawardena, Lannes, and Zarati proved that the Quillen homomorphism image is an isomorphism for G=Σn at p=2, but fails to be an isomorphism for odd primes. We prove that at odd primes, the restriction of the Quillen map to the subring of elements that are annihilated by all Steenrod operations that involve the Bockstein is an isomorphism for all n.