kernel of this map via Groebner basis

Question

There is a map $\phi$ from a polynomial ring $\mathbb{F}_2 [x,y,z]$ to $\mathbb{F}_4[t]$ where $\mathbb{F}_4=\{0,1,\omega,\omega^2\}$ is an algebraic extension of $\mathbb{F}_2=\{0,1\}$ with characteristic $2$. The map which is a ring homomorphism, is defined as follows $\phi: \mathbb{F}_2 [x,y,z] \rightarrow \mathbb{F}_4[t]$ such that $\phi:x\rightarrow 1+t, y\rightarrow 1+\omega t, z\rightarrow 1+\omega^2 t$. The Kernel of this map can be found by finding the Groebner basis for the ideal $K=<x-1-t,y-1-\omega t,z-1-\omega^2 t>$ and taking it's intersection with $\mathbb{F}_2 [x,y,z]$. (from Introduction to Groebner Basis by Adams and Loustaunau). Using this method, I get Kernel as $\{1- \omega+\omega y-z, 1-\omega+\omega x-y,x-2y+y^2+z-xz\}$ but the correct answer to the kernel is $\{1+x+y+z,1+xy+yz+zx\}$. What is the wrong step? Also, I calculated the Groebner basis for the ideal $K$ in Mathematica using DegreeRevLex monomial ordering with $t>x>y>z$.