This is a well known and well understood problem when the base field is $\mathbb C$. It was first studied by Chevalley and Weil (almost a century ago !) who where interested in modular curves (what else ?).

For a modern account, see

Kani, Ernst : The Galois-module structure of the space of holomorphic differentials of a curve. J. reine angew. Math. 367 (1986), 187-206 .

Nakajima, Shoichi : Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties. J. Number Theory 22, 115-123 (1986).

Here is a brief report. Warning : this problem is well understood for a tame action only, over an algebraically closed field $k$, in particular over $\mathbb C$, but it is still open, as far as I know, in the wild action case. So I assume tameness.

At each point $P\in X$, define the ramification character $\psi_P$ as the character given by the action of the cyclic stabilizer $G_P$ on the cotangent space $\mathcal m_P/\mathcal m_P^2$. Define the ramification module by

$$\Gamma _{G}=\sum_{P\in X}{\rm Ind}_{G_{P}}^{G}\sum_{l=1}^{e_{P}-1}l\psi _{P}^{l}$$

where $e_P$ is the ramification index at $P$. Then the equivariant Hurwitz formula says : there exists a unique $k\left[ G\right]$ -module $\widetilde{\Gamma} _{G}$ such that

$$\widetilde{\Gamma}_{G}^{\oplus \#G}=\Gamma _{G}$$ and in the Grothendieck ring $R_{k}(G)$ holds the following equality :

$$ \chi(G,\mathcal{O}_{X})=\chi(\mathcal{O}_{Y})\cdot \left[ k \left[ G\right] \right] -[ \widetilde{\Gamma} _{G}] $$

where, as you can assume, $\chi(G,\cdot)$ is the equivariant Euler characteristic, and $\chi(\cdot)$ is the ordinary Euler characteristic. From this, and Serre duality, which is of course equivariant, you get the structure of $H^0(X,\Omega^1_X)$.

The proof of the equivariant Hurwitz formula is not hard. Since it is an equality between characters, it is enough to prove it when you evaluate at each $g\in G$. For $g=e$, this is the usual Hurwitz formula. For $g\neq e$, one uses a Lefschetz fix point formula for the corresponding automorphism of $X$.

This is a well known and well understood problem when the base field is $\mathbb C$. It was first studied by Chevalley and Weil (almost a century ago !) who were interested in modular curves (what else ?).

For a modern account, see

Kani, Ernst : The Galois-module structure of the space of holomorphic differentials of a curve. J. reine angew. Math. 367 (1986), 187-206 .

Nakajima, Shoichi : Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties. J. Number Theory 22, 115-123 (1986).

Here is a brief report. Warning : this problem is well understood for a tame action only, over an algebraically closed field $k$, in particular over $\mathbb C$, but it is still open, as far as I know, in the wild action case. So I assume tameness.

At each point $P\in X$, define the ramification character $\psi_P$ as the character given by the action of the cyclic stabilizer $G_P$ on the cotangent space $\mathcal m_P/\mathcal m_P^2$. Define the ramification module by

$$\Gamma _{G}=\sum_{P\in X}{\rm Ind}_{G_{P}}^{G}\sum_{l=1}^{e_{P}-1}l\psi _{P}^{l}$$

where $e_P$ is the ramification index at $P$. Then the equivariant Hurwitz formula says : there exists a unique $k\left[ G\right]$ -module $\widetilde{\Gamma} _{G}$ such that

$$\widetilde{\Gamma}_{G}^{\oplus \#G}=\Gamma _{G}$$ and in the Grothendieck ring $R_{k}(G)$ holds the following equality :

$$ \chi(G,\mathcal{O}_{X})=\chi(\mathcal{O}_{Y})\cdot \left[ k \left[ G\right] \right] -[ \widetilde{\Gamma} _{G}] $$

where, as you can assume, $\chi(G,\cdot)$ is the equivariant Euler characteristic, and $\chi(\cdot)$ is the ordinary Euler characteristic. From this, and Serre duality, which is of course equivariant, you get the structure of $H^0(X,\Omega^1_X)$.

The proof of the equivariant Hurwitz formula is not hard. Since it is an equality between characters, it is enough to prove it when you evaluate at each $g\in G$. For $g=e$, this is the usual Hurwitz formula. For $g\neq e$, one uses a Lefschetz fix point formula for the corresponding automorphism of $X$.