  In the past few years, an action of $\mathrm{PGL}_2(\mathbb F_q)$ on the set
of irreducible polynomials in $\mathbb F_q[x]$ has been introduced and many
questions have been discussed, such as the characterization and number of
invariant elements. In this paper, we analyze some recent works on this action
and provide full generalizations of them, yielding final theoretical results on
the characterization and number of invariant elements.
