  We develop a complete obstruction theory for the $\mathbb{Z}_2$-index of a
compact connected 4-dimensional manifold with free involution. This
$\mathbb{Z}_2$-index, equal to the minimum integer $n$ for which there exists
an equivariant map with target the $n$-sphere with antipodal involution, is
computed in two steps using cohomology with twisted coefficients. The key
ingredient is a spectral sequence computing twisted cohomology of the orbit
space of a free involution on odd complex projective spaces. We illustrate the
main results with various examples including computation of the secondary
obstruction.
