Holomorphic differentials of alternating four covers

Suppose $k$ is an algebraically closed field of characteristic two, let $A_4$ be an alternating group on four letters, and let $H$ be the unique Sylow two-subgroup of $A_4$. Let $X$ be a smooth projective irreducible curve over $k$ with a faithful $A_4$-action such that the quotient curve $X/H$ is a projective line and the $H$-cover $X\to X/H$ is totally ramified, in the sense that it is ramified and every branch point is totally ramified. Under these assumptions, we determine the precise $kA_4$-module structure of the space of holomorphic differentials of $X$ over $k$. We show that there are infinitely many different isomorphism classes of indecomposable $kA_4$-modules that can occur as direct summands, and we give precise formulas for the multiplicities with which they occur.