Related to this question I also had some troubles to understand the classification of module categories over $Rep(G)$. Specifically, on page 12 of Ostrik's paper what is the category $\mathrm{Rep}^1(\tilde{H})$? The group $k^{\star}$ acting as "identity character on V" simply means $a.v=av$ for all $a \in k^*$ and $v \in V$? Then what is the structure of module category over $Rep(G)$? Tensor product should be after restricting representations of $G$ to $H$ and then inducing back to $\tilde{H}$?

Concretely, I was thinking about the following example. Let $N$ be a normal subgroup of $G$. Then $G$ acts on the irreducible representations of $N$. Let $\mathcal{O}$ be an orbit of this action and $\mathcal{M}$ be the full abelian subcategory of $Rep(N)$ generated by isomorphism classes of representations from $\mathcal{O}$. Then $\mathcal{M}$ is a module category over $Rep(G)$. What are the corresponding subgroup $H$ and cocyle $\omega \in H^2(H,\;k^*)$?