  We show that given a Frobenius algebra there is a unique notion of its second
quantization, which is the sum over all symmetric group quotients of n--th
tensor powers, where the quotients are given by symmetric group twisted
Frobenius algebras. To this end, we consider the setting of Frobenius algebras
given by functors from geometric categories whose objects are endowed with
geometric group actions and prove structural results, which in turn yield a
constructive realization in the case of n--th tensor powers and the natural
permutation action. We also show that naturally graded symmetric group twisted
Frobenius algebras have a unique algebra structure already determined by their
underlying additive data together with a choice of super--grading. Furthermore
we discuss several notions of discrete torsion andshow that indeed a
non--trivial discrete torsion leads to a non--trivial super structure on the
second quantization.
