  Let $K$ be a field, $\Gamma $ a finite group of field automorphisms of $K$,
$F$ the $\Gamma $-fixed field in $K$ and $G\leq $GL$_v(K)$ a finite matrix
group. Then the action of $\Gamma $ defines a grading on the symmetric algebra
of the $F$-space $K^v$ which we use to introduce the notion of homogeneous
$\Gamma $-conjugate invariants of $G$. We apply this new grading in invariant
theory to broaden the connection between codes and invariant theory by
introducing $\Gamma $-conjugate complete weight enumerators of codes. The main
result of this paper applies the theory from Nebe, Rains, Sloane to show that
under certain extra conditions these new weight enumerators generate the ring
of $\Gamma $-conjugate invariants of the associated Clifford-Weil groups. As an
immediate consequence we obtain a result by Bannai etal that the complex
conjugate weight enumerators generate the ring of complex conjugate invariants
of the complex Clifford group. Also the Schur-Weyl duality conjectured and
partly shown by Gross etal can be derived from our main result.
