Nontrivial irreducible component of permutation representation of projective general linear group of degree two on projective line

Let $$K$$ be a field. Consider the projective general linear group of degree two $$PGL(2,K)$$. This has a natural action on the projective line over $$K$$, i.e., the collection of one-dimensional subspaces of the two-dimensional vector space $$K^2$$. We thus get a permutation representation of $$PGL(2,K)$$ on the projective line $$\mathbb{P}^1(k)$$.

The action could be described in either of these ways:


 * For any element of $$PGL(2,K)$$, lift it to an element of $$GL(2,K)$$, and consider the image of any one-dimensional subspace under the element of $$GL(2,K)$$. Note that the image subspace does not depend on the choice of the lift, because any two lifts differ multiplicatively by a scalar matrix, which sends every subspace to itself.
 * Think of $$\mathbb{P}^1(K)$$ as $$K \cup \{ \infty\}$$. For an element of $$PGL(2,K)$$, consider a matrix $$\begin{pmatrix} a & b \\ c & d \\\end{pmatrix}$$ that is a lift of this element. The permutation induced by this is the map $$z \mapsto (az + b)/(cz + d)$$, where the value is taken to be $$\infty$$ if the denominator becomes $$0$$, and the image of $$\infty$$ is taken to be $$a/c$$ if $$c \ne 0$$ and to be $$\infty$$ if $$c = 0$$.

When $$K$$ is a finite field of size $$q$$, then this gives a permutation action of a finite group $$PGL(2,q)$$ on a finite set of size $$q + 1$$. View this as a linear representation in any characteristic not dividing the order of $$PGL(2,q)$$. This linear representation splits as a direct sum of a trivial representation and a nontrivial irreducible representation of degree $$q + 1 - 1 = q$$. Our goal here is to discuss this irreducible component.