Let $R$ be a commutative ring with identity. Let $I,J$ be ideals of $R$, such that $I\subseteq J$. Then the canonical map $\Phi: R/I\rightarrow R/J$ defined as $\Phi(I+x)=J+x$ is a surjective ring homomorphism.
Now take $R=\mathbb{Z}$ and $I=m\mathbb{Z}$ and $J=n\mathbb{Z}$ where $n\mid m$. It is clear that the image of ${(\mathbb{Z}/m\mathbb{Z})}^{\times}$ under $\Phi$ is contained in ${(\mathbb{Z}/n\mathbb{Z})}^{\times}$. If we denote $\phi$ as the restriction of $\Phi$ to ${(\mathbb{Z}/m\mathbb{Z})}^{\times}$ then we have
$$\phi:{(\mathbb{Z}/m\mathbb{Z})}^{\times}\rightarrow {(\mathbb{Z}/n\mathbb{Z})}^{\times}$$
I checked a few cases and it seems $\phi$ is surjective. Is it true in general ?
