We establish a twisted analog of our recent work on vertex
representations and the McKay correspondence. For each finite group
$\Gamma$ and a virtual character of $\Gamma$, we construct twisted
vertex operators on the Fock space spanned by the super spin
characters of the spin wreath products $\Gamma\wr \tilde {S}_n$ of
$\Gamma$ and a double cover of the symmetric group $S_n$ for all
$n$. When $\Gamma$ is a subgroup of ${\rm SL}_2(\mathbb {C})$ with
the McKay virtual character, our construction gives a group-theoretic
realization of the basic representations of the twisted affine and
twisted toroidal Lie algebras. When $\Gamma$ is an arbitrary finite
group and the virtual character is trivial, our vertex operator
construction yields the spin character tables for $\Gamma\wr \tilde
{S}_n$.