Moduli space of nonnegatively curved metrics on manifolds of dimension $4k+1$

In each dimension $4k+1\ge 9$, we exhibit infinite families of closed manifolds with fundamental group ${\mathbb{Z}}_2$ for which the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with this property were known before only in dimension $5$ and dimensions $4k+3\ge 7$.