Suppose we have a sufficiently nice (example below) pathwise connected topological space $X$ with a double cover $\pi : \widetilde X \to X$. There is an associated free $\mathbb Z_2$-action on $\widetilde X$ interchanging the sheets of the cover; denote by $\iota : \widetilde X \to \widetilde X$ the map corresponding to the nonzero element of $\mathbb Z_2$. The question is: under which conditions is $\iota$ detectable by its action on the homology of $\widetilde X$, $\iota_* : H_*(\widetilde X) \to H_*(\widetilde X)$? In other words, when is $\iota_*$ not the identity? And when it is not, how can one compute the $\pm 1$-eigenspaces of $\iota_*$?

(The coefficients may be freely chosen to give the "best" answer possible.)

Examples:

1) If $\pi$ is the trivial covering, $\widetilde X$ is the disjoint union of two copies of $X$ and therefore $H_*(\widetilde X) = H_*(X) \oplus H_*(X)$. The action of $\iota_*$ is by interchanging the summands and therefore it is detectable.

2) If $X = \widetilde X = S^1$ and $\pi$ is the unique nontrivial double covering, then $\iota$ is isotopic to the identity on $S^1$ and therefore acts trivially on $H_*(S^1)$.

3) If $X = \mathbb R P^2$, $\widetilde X = S^2$ and $\pi : \widetilde X \to X$ is the orientation covering, then $\iota$ acts as the antipodal map on $S^2$. The map $\iota_*$ acts trivially on $H_0(S^2)$ but on $H_2(S^2;\mathbb Z) = \mathbb Z$, for example, it acts as the multiplication by $-1$.

4) The real example I have in mind is as follows. Let $M$ be a smooth manifold, $L \subset M$ a smooth submanifold, and assume for simplicity both are closed and connected. Fix a point $x_0 \in L$, and let $X$ be the space of smooth maps $(D^2, \partial D^2, 1) \to (M,L,x_0)$, representing $0 \in \pi_2(M,L,x_0)$. Here $D^2 \subset \mathbb C$ is the closed unit disk. There is an isomorphism $\pi_1(X,x_0) \simeq \pi_3(M,L,x_0)$ ($x_0$ being the constant map, of course). Suppose $\widetilde X$ is the double cover corresponding to a real line bundle whose first Stiefel-Whitney class is given by a homomorphism $w : \pi_3(M,L,x_0) \to \mathbb Z_2$ (in view of the above isomorphism this is equivalent to specifying a class in $H^1(X;\mathbb Z_2)$). What can be said about this example?