gr.group theory - Module categories over $Rep(G)$.

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 $mathrm{Rep}^1(tilde{H})$? $k^*$ acting as "identity character on V" 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 $H$ be a subgroup of $G$. Then $Rep(H)$ is a module category over $Rep(G)$ via tensor product as $H$-modules. What is the decomposition of $Rep(H)$ in indecomposable module categories and what are the corresponding subgroups $H$ and cocyles $omega in H^2(H,;k^*)$ for each indecomposable subcategory?