The Gromov norm of a compact oriented n-manifold is a norm on the homology (with real coefficients) given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle. The Gromov norm of the manifold is the Gromov norm of the fundamental class. It is known, the following class of manifold has a non-zero Gromov norm:
$(1)$ Oriented closed connected Riemannian manifolds of negative sectional curvature.
$(2)$ Oriented closed connected hyperbolic manifolds.
$\textbf{Q})$ I am interested in knowing a list of examples of orientable manifolds with non-zero Gromov norm that admit a free $\mathbb{Z}_2$-action.
As one example, we can consider the genus $g>1$ torus $T_g$. Since $T_g$ is a hyperbolic manifold, its Gromov norm is non-zero by (2). Moreover, If we assume this is embedded in $\mathbb{R}^3$ in a standard way, with the “center” of the torus at the origin, then the antipodal map $x\to -x$ define a free $\mathbb{Z}_2$-action on it.