Homomorphism of quasianalytic local rings

Let $\mathcal{C}_n$ be a local quasi-analytic subring of the ring of germs of $C^\infty$ functions on $\mathbb{R}^n$, and let $\mathcal{C}=\{ \mathcal{C}_n , n\in\mathbb{N}\}$. We suppose that $\mathcal{C}$ is closed under composition. Consider a map $\varphi : (\mathbb{R}^n, 0)\rightarrow (\mathbb{R}^k, 0)$ vanishing at zero, where $\varphi$ is a $k$-tuple $(\varphi_1,\ldots,\varphi_k)$ and $\varphi_1,\ldots,\varphi_k$ are in $\mathcal{C}_n$. Then $\varphi$ defines uniquely a map $\phi: \mathcal{C}_k \rightarrow \mathcal{C}_n$ by composition, and $\phi$ induces a morphism $\hat{\phi}: \hat{\mathcal{C}_k }\rightarrow \hat{\mathcal{C}_n}$ between completions. We let $\phi_* : \frac{\hat{\mathcal{C}_k} }{\mathcal{C}_k }\rightarrow\frac{\hat{\mathcal{C}_n }}{\mathcal{C}_n }$ be the homomorphism of groups induced by $\phi$ and $\hat{\phi}$ in the obvious manner. In the analytic case, i.e. when each $\mathcal{C}_n$ is the ring of germs of real analytic functions, M. Eakin and A. Harris give a condition under which $\phi_*$ is injective. In this paper we prove that the same statement does not hold for a quasianalytic system unless this system is analytic.