  Let $n$ be a positive integer and let $\mathbb F_{q^n}$ be the finite field
with $q^n$ elements, where $q$ is a power of a prime. This paper introduces a
natural action of the Projective Semilinear Group $\text{P}\Gamma \text{L}(2,
q^n)=\text{PGL}(2, q^n)\rtimes \text{Gal}(\mathbb{F}_{q^n}/\mathbb{F}_q)$ on
the set of monic irreducible polynomials over the finite field
$\mathbb{F}_{q^n}$. Our main results provide information on the
characterization and number of fixed points.
