  Goppa, in the 1970s, discovered the relation between algebraic geometry and
codes, which led to the family of Goppa codes. As one of the most interesting
subclasses of linear codes, the family of Goppa codes is often chosen as a key
in the McEliece cryptosystem. Knowledge of the number of inequivalent binary
Goppa codes for fixed parameters may facilitate in the evaluation of the
security of such a cryptosystem. Let $n\geq5$ be an odd prime number, let
$q=2^n$ and let $r\geq3$ be a positive integer satisfying $\gcd(r,n)=1$. The
purpose of this paper is to establish an upper bound on the number of
inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree
$r$.A potential mathematical object for this purpose is to count the number of
orbits of the projective semi-linear group ${\rm
PGL}_2(\mathbb{F}_q)\rtimes{\rm Gal}(\mathbb{F}_{q^r}/\mathbb{F}_2)$ on the set
$\mathcal{I}_r$ of all monic irreducible polynomials of degree $r$ over the
finite field $\mathbb{F}_q$. An explicit formula for the number of orbits of
${\rm PGL}_2(\mathbb{F}_q)\rtimes{\rm Gal}(\mathbb{F}_{q^r}/\mathbb{F}_2)$ on
$\mathcal{I}_r$ is given, and consequently, an upper bound for the number of
inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree
$r$ is derived. Our main result naturally contains the main results of Ryan
(IEEE-TIT 2015), Huang and Yue (IEEE-TIT, 2022) and, Chen and Zhang (IEEE-TIT,
2022), which considered the cases $r=4$, $r=6$ and $\gcd(r,q^3-q)=1$
respectively.
