Riemannian holonomy of a covering

${\rm Hol}(Y)$ is indeed a subgroup of ${\rm Hol}(X)$. The fundamental group introduces connected components to the holonomy; that is, there is a surjection $\pi_1(X)\to{\rm Hol}(X)/{\rm Hol}(X)^\circ={\rm Hol}(X)/{\rm Hol}(\tilde{X})$ (with $\tilde{X}$ being the universal cover). I'm not aware that this can be made more precise in general (if you read German, have a look at Helga Baum's book "Eichfeldtheorie", Section 5.1; can't think of a good English reference now, but something on this is probably in Kobayashi & Nomizu, Book I).

Example: For a complete flat manifold $X$, the fundamental group is a group of affine transformations on $\tilde{X}=\mathbb{R}^n$. The (linear) holonomy group is then given by the projection to the linear parts of the fundamental group (e.g. Joe Wolf's book "Spaces of constant curvature", Chapter 3).

Hope this helped.