This paper studies the factorization of polynomials of the form \(F_r(x) = bx^{q^{r}+1}-ax^{q^r} + dx-c\) over a finite field \(\mathbb{F}_q\). The authors show that all the irreducible factors of \(F_r(x)\) are exactly those polynomials that are invariant under a natural action of a nontrivial element \([A]\) of \(\mathrm{PGL}(2, q)\) where \[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}. \] Here the equivalence relation on \(\mathrm{GL}(2, q)\) is defined by \(A\sim B\) if and only if \(B= \lambda A\) for some nonzero \(\lambda \in\mathbb{F}_q\). The action of an equivalence class \([A]\) on a monic irreducible polynomial \(f\) of degree \(n\) over \(\mathbb{F}_q\) is defined by \([A] \circ f := the~unique~monic~polynomial~ g~with ~ g \sim~ A\circ f\), where \( (A\circ f)(x) := (bx+d)^n f\left( \frac{ax+c}{bx+d} \right)\) and two polynomials \(f~\sim g\) if and only if \(g = \lambda f\) for some nonzero \(\lambda \in \mathbb{F}_q\). As a consequence, the authors obtain some asymptotic results on the degrees of irreducible factors of \(F_r(x)\) and the number of \([A]\)-invariant irreducible polynomials. Namely, let \(D = ord[A]\), almost all irreducible factors of \(F_r(x)\) have degree \(Dr\), for large values of \(r\) and \(D \geq 2\). Also the number of monic \([A]\)-invariant irreducible polynomials of degree \(Dn\) is asymptotic equal to \(\phi(D)\frac{q^n}{Dn}\) as \(n\) goes to infinity, where \(\phi\) is the Euler's phi-function and \(D \geq 1\). The latter result generalizes earlier results on self-reciprocal polynomials by \textit{H. Meyn} [Appl. Algebra Eng. Commun. Comput. 1, 43--53 (1990; Zbl 0724.11062)] and alternate polynomials by \textit{J. F. Michon} and \textit{P. Ravache} [Finite Fields Appl. 16, No. 3, 163--174 (2010; Zbl 1206.11146)].