Use Z/(2)[a,g,r,z];

K:=Ideal(a^2+a, g^2+g, r^2+r, z^2+z);

I:=Ideal((a+1)(g+1)(r+1)(z+1), (g+1)(r+1)a, g(r+1)(z+1), (g+1)r, zg, r(z+1)) + K;

ReducedGBasis(I);