  Given a commutative local ring $(R,\mathfrak m)$ and an ideal $I$ of $R$, a
family of quotients of the Rees algebra $R[It]$ has been recently studied as a
unified approach to the Nagata's idealization and the amalgamated duplication
and as a way to construct interesting examples, especially integral domains.
When $R$ is noetherian of prime characteristic, we compute the Hilbert-Kunz
function of the members of this family and, provided that either $I$ is
$\mathfrak{m}$-primary or $R$ is regular and F-finite, we also find their
Hilbert-Kunz multiplicity. Some consequences and examples are explored.
