I have a question about an argument in the proof of Lemma 1.2.(1) in Quotients of K3 surfaces modulo involutions by D. Q. Zhang:

Let Let $(X, \sigma)$ be X be a smooth projective K3 surface with an involution $\sigma$ such that $\sigma^*\omega=-\omega$ for a non-zero holomorphic $2$-form $\omega$. Let $S := X/\sigma$ be the quotient space and $f : X \to S$ the quotient morphism.

Then (1) $S$ is a smooth surface with irregularity $q(S):= h^1(S, \mathcal{O}_S) = 0$.

The first part on smoothness is fine, see prev. Lemma 1.1.(1). But how do we obtain that the irregularity $q(S)$ is zero?

It is well known that irregularity of K3 surfaces is zero, and we have a dominant finite morphism $f: X \to S$.

Question: Is it always true that $q(S) \le q(X)$? How to see it?

Remark: Zhang works in complex setting, ie base field algebraically closed of char $0$. Does it matter for the statement on proposed inequality between irregularities?