Projective general linear group of degree two over a prime field is complete

Statement
Let $$p$$ be a prime number. The fact about::projective general linear group of degree two over the prime field $$\mathbb{F}_p$$, i.e., the group $$PGL(2,p)$$, is a fact about::complete group: it is a centerless group and every automorphism of it is inner.

Note that $$PGL(2,q)$$ is not complete when $$q$$ is a prime power that is not itself a prime -- there are automorphisms of the group arising from Galois automorphisms of the field extension $$\mathbb{F}_q$$ over its prime subfield.