  Kreck's modified surgery theory reduces the classification of closed,
connected 4-manifolds, up to connect sum with some number of copies of
$S^2\times S^2$, to a series of bordism questions. We implement this in the
case of unorientable 4-manifolds M and show that for some choices of
fundamental groups, the computations simplify considerably. We use this to
solve some cases in which $\pi_1(M)$ is finite of order 2 mod 4: under an
assumption on cohomology, there are nine stable diffeomorphism classes for
which M is pin$^+$, one stable diffeomorphism class for which M is pin$^-$, and
four stable diffeomorphism classes for which M is neither. We also determine
the corresponding stable homeomorphism classes.
