  A knot $K$ in $S^3$ is $q$-periodic if it admits a symmetry that is conjugate
to a rotation of order $q$ of $S^3$. If $K$ admits a symmetry which is a
homeomorphism without fixed point of period $q$ of $S^3$, then $K$ is called
freely $q$-periodic.
  In a previous paper, we obtained, as a consequence of Flyping Theorem due to
Menasco and Thislethwaite, that the $q$-periodicity with $q>2$ can be
visualized in an alternating projection as a rotation of the projection sphere.
  In this paper, we show that the free $q$-action of an alternating knot can be
represented on some alternating projection as a composition of a rotation of
order $q$ with some flypes all occurring on the same twisted band diagram of
its essential Conway decomposition. Therefore, for an alternating knot to be
freely periodic, its essential decomposition must satisfy certain conditions.
We show that any free or non-free $q$-action is somehow visible (virtually
visible) and give some sufficient criteria to detect the existence of
$q$-actions from virtually visible projections.
  Finally, we show how the Murasugi decomposition into atoms enables us to
determine the visibility type $(q,r)$ of the freely $q$-periodic alternating
knots ($(q,r)$-lens knots); in fact, we only need to focus on a certain atom of
their Murasugi decomposition to deduce their visibility type.
