  Let $X$ be a smooth projective geometrically irreducible curve over a perfect
field $k$ of positive characteristic $p$. Suppose $G$ is a finite group acting
faithfully on $X$ such that $G$ has non-trivial cyclic Sylow $p$-subgroups. We
show that the decomposition of the space of holomorphic differentials of $X$
into a direct sum of indecomposable $k[G]$-modules is uniquely determined by
the lower ramification groups and the fundamental characters of closed points
of $X$ that are ramified in the cover $X\to X/G$. We apply our method to
determine the $\mathrm{PSL}(2,\mathbb{F}_\ell)$-module structure of the space
of holomorphic differentials of the reduction of the modular curve
$\mathcal{X}(\ell)$ modulo $p$ when $p$ and $\ell$ are distinct odd primes and
the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on this reduction is not tamely
ramified. This provides some non-trivial congruences modulo appropriate maximal
ideals containing $p$ between modular forms arising from isotypic components
with respect to the action of $\mathrm{PSL}(2,\mathbb{F}_\ell)$ on
$\mathcal{X}(\ell)$.
