Fix a point $p$ in the torus $S^{1} \times S^{1}$. Let $G$ be the group of homeomorphisms $f$ from the torus $S^{1} \times S^{1}$ to itself such that $f(p)=p$. Determine a non-trivial homomorphism $\phi$ from $G$ to the group $\operatorname{GL}(2, \mathbb{Z})$.

[The group $\mathrm{GL}(2, \mathbb{Z})$ consists of $2 \times 2$ matrices with integer coefficients that have an inverse which also has integer coefficients.]

Establish whether each $f$ in the kernel of $\phi$ is homotopic to the identity map.