  Let $\mathfrak{X}$ be a class of finite groups closed under taking subgroups,
homomorphic images and extensions. It is known that if $A$ is a normal subgroup
of a finite group $G$ then the image of an $\mathfrak{X}$-maximal subgroup $H$
of $G$ in $G/A$ is not, in general, $\mathfrak{X}$-maximal in $G/A$. We say
that the reduction $\mathfrak{X}$-theorem holds for a finite group $A$ if, for
every finite group $G$ that is an extension of $A$ (i. e. contains $A$ as a
normal subgroup), the number of conjugacy classes of $\mathfrak{X}$-maximal
subgroups in $G$ and $G/A$ is the same. The reduction $\mathfrak{X}$-theorem
for $A$ implies that $HA/A$ is $\mathfrak{X}$-maximal in $G/A$ for every
extension $G$ of $A$ and every $\mathfrak{X}$-maximal subgroup $H$ of $G$. In
this paper, we prove that the reduction $\mathfrak{X}$-theorem holds for $A$ if
and only if all $\mathfrak{X}$-maximal subgroups are conjugate in $A$ and
classify the finite groups with this property in terms of composition factors.
