Let X be a closed, oriented, Riemannian 4-manifold with ${{b_2}^+}(x);>;1$ and of simple type. Suppose that ${sigma};:;X;{ ightarrow};X$ is an involution preserving orientation with an oriented, connected, compact 2-dimensional submanifold $Sigma$ as a fixed point set with ${SigmacdotSigma};{geq};0;and;[Sigma];{ eq};0;{in};H_2(X;mathbb{Z})$. We show that if _X(Sigma);+;{SigmacdotsSigma};{ eq};0$ then the $Spin^{C}$ bundle $={P}$ is not $mathbb{Z}_2-equivariant$, where det $={P};=;L$ is a basic class with $c_1(L)[Sigma];=;0$.