  A. Takahashi suggested a conjectural method to find mirror symmetric pairs
consisting of invertible polynomials and symmetry groups generated by some
diagonal symmetries and some permutations of variables. Here we generalize the
Saito duality between Burnside rings to a case of non-abelian groups and prove
a "non-abelian" generalization of the statement about the equivariant Saito
duality property for invertible polynomials. It turns out that the statement
holds only under a special condition on the action of the subgroup of the
permutation group called here PC ("parity condition"). An inspection of data on
Calabi-Yau threefolds obtained from quotients by non-abelian groups shows that
the pairs found on the basis of the method of Takahashi have symmetric pairs of
Hodge numbers if and only if they satisfy PC.
