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