  The stabilizer formalism for quantum error-correcting codes has been, without
doubt, the most successful at producing examples of quantum codes with strong
error-correcting properties. In this paper, we discuss strong automorphism
groups of stabilizer codes, beginning with the analogous notion from the theory
of classical codes. Two weakenings of this concept, the weak automorphism group
and Clifford-twisted automorphism group, are also discussed, along with many
examples highlighting the possible relationships between the types of
"automorphism groups". In particular, we construct an example of a $[[10,0,4]]$
stabilizer code showing how the Clifford-twisted automorphism groups might be
connected to the Mathieu groups. Finally, nonexistence results are proved
regarding stabilizer codes with highly transitive strong and weak automorphism
groups, suggesting a potential inverse relationship between the
error-correcting properties of a quantum code and the transitivity of those
automorphism groups.
