  We show that a free action $G \curvearrowright X$ is almost finite if its
restriction to some infinite normal subgroup of $G$ is almost finite. Consider
the class of groups which contains all infinite groups of locally
subexponential growth and is closed under taking inductive limits and
extensions on the right by any amenable group. It follows that all free actions
of a group from this class on finite-dimensional spaces are almost finite and
therefore that minimal such actions give rise to classifiable crossed products.
In particular, that gives a much easier proof for the recent result of Kerr and
the author on elementary amenable groups.
