  We give a complete classification of topological field theories with
reflection structure and spin-statistics in one and two spacetime dimensions.
Our answers can be naturally expressed in terms of an internal fermionic
symmetry group $G$ which is different from the spacetime structure group.
Fermionic groups encode symmetries of systems with fermions and time reversing
symmetries. We show that 1-dimensional topological field theories with
reflection structure and spin-statistics are classified by finite dimensional
hermitian representations of $G$. In spacetime dimension two we give a
classification in terms strongly $G$-graded stellar Frobenius algebras. Our
proofs are based on the cobordism hypothesis. Along the way, we develop some
useful tools for the computation of homotopy fixed points of 2-group actions on
bicategories.
