Invariant theory of a special group action on irreducible polynomials over finite fields

In the past few years, an action of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2({\mathbb{F}}_q)$ 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.