Stable diffeomorphism classification of some unorientable 4-manifolds

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.