  Let $\mathfrak{H}$ be a Fitting class and $\mathfrak{F}$ a formation. We call
a subgroup $\mathcal{N}_{\mathfrak{H},\mathfrak{F}}(G)$ of a finite group $G$
the $\mathfrak{H}$-$\mathfrak{F}$-norm of $G$ if
$\mathcal{N}_{\mathfrak{H},\mathfrak{F}}(G)$ is the intersection of the
normalizers of the products of the $\mathfrak{F}$-residuals of all subgroups of
$G$ and the $\mathfrak{H}$-radical of $G$. Let $\pi$ denote a set of primes and
let $\mathfrak{G}_\pi$ denote the class of all finite $\pi$-groups. We call the
subgroup $\mathcal{N}_{\mathfrak{G}_\pi,\mathfrak{F}}(G)$ of $G$ the
$\pi\mathfrak{F}$-norm of $G$. A normal subgroup $N$ of $G$ is called
$\pi\mathfrak{F}$-hypercentral in $G$ if either $N=1$ or $N>1$ and every
$G$-chief factor below $N$ of order divisible by at least one prime in $\pi$ is
$\mathfrak{F}$-central in $G$. Let $Z_{\pi\mathfrak{F}}(G)$ denote the
$\pi\mathfrak{F}$-hypercentre of $G$, that is, the product of all
$\pi\mathfrak{F}$-hypercentral normal subgroups of $G$. In this paper, we study
the properties of the $\mathfrak{H}$-$\mathfrak{F}$-norm, especially of the
$\pi\mathfrak{F}$-norm of a finite group $G$. In particular, we investigate the
relationship between the $\pi'\mathfrak{F}$-norm and the
$\pi\mathfrak{F}$-hypercentre of $G$.
