This is a follow up to this question.
Let $(R,+,\cdot)$ be a finite ring.
Definition Given the dual group $\widehat{R}$ of $(R,+)$, a character $\chi\in\widehat{R}$ is said to be primitive with respect to a collection $\mathcal{C}$ of subgroups of $(R,+)$, if $\chi$ does not descend to a character on any quotient group $R/S$ for all $S\in \mathcal{C}$. To descend here is canonical in the sense that $\chi$ is invariant on the elements of any coset of $R/S$.
My question is the following problem:
Are there any generic non-trivial conditions on $\mathcal{C}$ such that $$\#\{\text{primitive characters in $\widehat{R}$ w.r.t $\mathcal{C}$}\}=\#R^\times,$$ i.e. when does the number of primitive* characters of the abelian group $(R,+)$ w.r.t. $\mathcal{C}$ equal the number of units in $R$?
Qiaochu Yuan's answer here shows that $\mathbb{Z}/n\mathbb{Z}$ is the only example when one considers all proper subquotients. My question is whether a well-defined non-trivial collection of, rather than all proper, quotients leads to an affirmation of the equality above.
