Cohomological non-rigidity of generalized real Bott manifolds of height 2

We investigate the following problem: When do two generalized real Bott manifolds of height 2 have isomorphic cohomology rings with $\mathbb Z/2$ coefficients and also when are they diffeomorphic? It turns out that in general cohomology rings with $\mathbb Z/2$ coefficients do not distinguish those manifolds up to diffeomorphism. This gives a negative answer to the cohomological rigidity problem for real toric manifolds posed earlier by Y. Kamishima and the present author. We also prove that generalized real Bott manifolds of height 2 are diffeomorphic if they are homotopy equivalent.