Question 1. Let $X_i$ be an infinite family of closed, orientable, smooth 4-manifolds with the following properties:

a) $\pi_1(X_i) = \mathbb{Z}\times \mathbb{Z_{2}}$ for any $i = 1, 2, \cdots $

b) all the homology groups of $X_i$ and $X_j$ with integer coefficients are same

Is it true in this family there are infinitely many homeomorphic 4-manifolds?

Does this follow from Freedman's classification theorem since $\mathbb{Z}\times \mathbb{Z_{2}}$ is a "good" group?

Question 2. What if $X_i$ has a boundary? How much is known in non simply-connected case?