Let $G$ be a finite abelian group and $H$ a subgroup if $G$. If we denote $\mathcal{C}(G)$ to the set of maps $\phi:G\to \mathbb{C}$ which are constant in a conjugation class. If $(-)_H:\mathcal{C}(G)\rightarrow\mathcal{C}(H)$ is the restriction function to $H$ show that $(-)_H$ is surjective and identify its kernel.
My attempt: Since $G$ is abelian, every conjugation class has only one element, so if $\phi\in \mathcal{C}(H)$ then by giving if ramdom values in the rest of the elements of $G$ we have the surjectivity. The kernel is giving by the functions which are $0$ in $H$. Am I right? Is there a better interpretation in the context of representation theory?
You're right.
A bit representation theoretic way to prove this — I wouldn't say this is a better interpretation — is showing a section map explicitly, i.e., proving $$\left(\frac{1}{\lvert G : H \rvert}\theta^G\right)_H = \theta$$ for every class function $\theta$ of $H$. This is just a degenerated identity of the Mackey decomposition.