I have a question about an argument in the proof of *Lemma 1.2.(1)* in [Quotients of K3 Surfaces Modulo Involutions](https://arxiv.org/abs/math/9905193) 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 irregilatity 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? 

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