  This paper investigates the relation between linear codes and the stabilizer
in ${\rm GL}_2(\mathbb{C})$ of their weight enumerators. We prove a result on
the finiteness of stabilizers and give a complete classification of linear
codes with infinite stabilizer in the non-binary case. We present an efficient
algorithm to compute explicitly the stabilizer of weight enumerators and we
apply it to the family of Reed-Muller codes to show that some of their weight
enumerators have trivial stabilizer.
