  Classical Goppa codes are a well-known class of codes with applications in
code-based cryptography, which are a special case of alternant codes. Many
papers are devoted to the search for Goppa codes with a cyclic extension or
with a cyclic parity-check subcode. Let $\Bbb F_q$ be a finite field with
$q=2^l$ elements, where $l$ is a positive integer. In this paper, we determine
all the generator polynomials of cyclic expurgated or extended Goppa codes
under some prescribed permutations induced by the projective general linear
automorphism $A \in PGL_2(\Bbb F_q)$. Moreover, we provide some examples to
support our findings.
