  We present the unoriented version of the Schur and Bogomolov multiplier
associated with a finite group $G$. We show that the unoriented Schur
multiplier is isomorphic to the second cohomology group $H^2(G;\mathbb{F}_2)$.
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. Disregarding the subgroup
generated by the trivial $G$-bundle over the projective space, we show that the
unoriented Bogomolov multiplier is a complete obstruction. This means that a
free surface action over an unoriented surface equivariantly bounds if and only
if the class in the unoriented Bogomolov multiplier is trivial. We show that
the unoriented Bogomolov multiplier is trivial for abelian, dihedral, and
symmetric groups. Because the group cohomology $H^2(G;\mathbb{F}_2)$ is trivial
for a group of odd order, there are plenty of examples where the usual
Bogomolov multiplier is not trivial while the unoriented Bogomolov multiplier
is trivial. However, we show that there is a group of order $64$ where the
unoriented Bogomolov multiplier is non-trivial.
