  Let $\mathcal{C}$ be a finite tensor category, and let $\mathcal{M}$ be an
exact left $\mathcal{C}$-module category. The relative Serre functor of
$\mathcal{M}$ is an endofunctor $\mathbb{S}$ on $\mathcal{M}$ together with a
natural isomorphism $\underline{\mathrm{Hom}}(M, N)^* \cong
\underline{\mathrm{Hom}}(N, \mathbb{S}(M))$ for $M, N \in \mathcal{M}$, where
$\underline{\mathrm{Hom}}$ is the internal Hom functor of $\mathcal{M}$. In
this paper, we discuss the case where $\mathcal{C}$ and $\mathcal{M}$ are the
category of modules over a finite-dimensional Hopf algebra $H$ and the category
of modules over an $H$-comodule algebra $L$, respectively. We give an explicit
description of the relative Serre functor of $\mathcal{M}$ and its twisted
module structure in terms of the Frobenius structure of $L$. We also study
pivotal structures on $\mathcal{M}$ and give some concrete examples.
