Action of covering involution on Neron-Severi group

Consider a normal variety $$S$$, and a ramified double cover $$f:\tilde{S}\to S$$, $$\tilde{S}$$ smooth, such that $$f$$ is étale over the nonsingular locus of $$S$$. Call $$j$$ the covering involution.

This is the most complete reference I could find: https://arxiv.org/pdf/1803.00799.pdf , Section $$2$$, but in general I hardly find references about ramification coverings. Here it is said that the structure sheaf of the covering involution may be written as $$\mathcal{O}_X\oplus \mathcal{R}$$, where $$\mathcal{R}$$ is a certain reflexive sheaf on $$X$$, see Lemma $$2.6$$.

The covering involution acts as the identity on the pullback of the structure sheaf of $$X$$, and as minus the identity on $$\mathcal{R}$$. This leads me to think that the involution $$j$$ acts as the identity on a class $$D$$ in the Néron-Severi group of $$\tilde{S}$$ if and only if $$D$$ can be written as the pullback through $$f$$ of a class in the Néron-Severi group of $$S$$. Anyway I couldn't figure out how to prove it.

Is it correct? Thank you!

No, this is not correct. Take for example an etale double covering $$C' \to C$$ of smooth projective curves. Then $$NS(C') = \mathbb{Z}$$ via the degree map, and similarly $$NS(C) = \mathbb{Z}$$. Furthermore, the involution acts on $$NS(C')$$ trivially. But the image of the pullback map $$NS(C) \to NS(C')$$ is the subgroup $$2\mathbb{Z} \subset \mathbb{Z} = NS(C')$$ of index 2.