Based on the question Question about the totient function and congruence classes, I would like to ask two new questions:
Question 1
Does it hold that if $k\mid\varphi(n)$ then the elements of the multiplicative group $\newcommand{\Z}{\mathbb Z}(\Z/n\Z)^*$ maps via $\psi_k$ defined by $\psi_k(m)=m\mbox{ mod }k$ into a subset of $\Z/k\Z$ in a way so that the pre-images of each $\alpha\in\mbox{Im}(\psi_k)$ have the same size.
Question 2
Which additional conditions must $k,n$ where $k\mid\varphi(n)$ satisfy in order for $\psi_k$ defined in the previous question to be surjective?
In the original question you can see that user11977 points out that $\psi_2$ is not surjective for $n=12$.
My thoughts, the Carmichael function
So far my thoughts are that this has to do with the Carmichael function and the product of cyclic groups connected to the multiplicative group of integers $(\Z/n\Z)^*$. (see this link)