  Given a $p$-group $G$ and a subgroup-closed class $\mathfrak{X}$, we
associate with each $\mathfrak{X}$-subgroup $H$ certain quantities which count
$\mathfrak{X}$-subgroups containing $H$ subject to further properties. We show
in Theorem I that each one of the said quantities is always $\equiv 1 \pmod p$
if and only if the same holds for the others. In Theorem II we supplement the
above result by focusing on normal $\mathfrak{X}$-subgroups and in Theorem III
we obtain a sharpened version of a celebrated theorem of Burnside relative to
the class of abelian groups of bounded exponent. Various other corollaries are
also presented.
