  Let $H$ be a subnormal co-compact closed subgroup of a Hausdorff topological
group $T$ and $X$ a compact Hausdorff space. We prove the inheritance theorem:
A point of $X$ is almost periodic (a.p.) for $T\curvearrowright X$ iff it is
a.p. for $H\curvearrowright X$. Moreover, if $T\curvearrowright X$ is minimal
with $H\lhd T$, then $\mathscr{O}_H\colon X\rightarrow2^X$,
${x\mapsto\overline{Hx}}$ is a continuous mapping, and, $T\curvearrowright X/H$
is an a.p. nontrivial factor of $T\curvearrowright X$ iff $T\curvearrowright
X\times T/H$ is not minimal.
