Let $B$ and $C$ be complex constants. Additionally, let $x,y,y,$ and $z$ be complex variables. What are the automorphisms of the quartic equation $((C-2)B-2C-12)(w^4+x^4+y^4+z^4)+((2(C-2))B^2+(2C^2+8C-24)B-4C^2-24C-64)xyzw+(-(C-2)B^2+(2C+12)B)(w^2x^2+y^2z^2)+((-2C-12)B-12C-40)(w^2z^2+x^2y^2)+((-C^2+2C)B+2C^2+12C)(w^2y^2+x^2z^2)=0$
Obviously, the quartic is left invariant under permutations such as $[w:x:y:z]\rightarrow[x:w:z:y]$ and when we multiply two variables by negative one. Is there an additional automorphism?
Update: I proved that there does exist an additional automorphism. However, I have not yet been able to determine what it is.