Extending free actions of finite groups on unoriented surfaces

We present the unoriented versions of the Schur and Bogomolov multipliers associated with a finite group $G$. We show that the unoriented Schur multiplier is isomorphic to the second cohomology group $H^2(G;{\text{ZZ}}_2)$. We define the unoriented Bogomolov multiplier as the quotient of the unoriented Schur multiplier by the subgroup generated by classes over the disjoint union of tori, Klein bottles, and projective spaces. We prove that the unoriented Bogomolov multiplier is trivial for abelian, dihedral, symmetric, and alternating groups. Since $H^2(G;{\text{ZZ}}_2)$ is trivial for any group of odd order, there are numerous examples where the classical Bogomolov multiplier is nontrivial while its unoriented counterpart is trivial. Nevertheless, we exhibit a group of order $64$ for which the unoriented Bogomolov multiplier is nontrivial.