Related to this [question](https://mathoverflow.net/questions/3309/are-there-two-groups-which-are-categorically-morita-equivalent-but-only-one-of-wh) I also had some troubles to understand the classification of module categories over $Rep(G)$. Specifically, on page 12 of Ostrik's [paper](http://arxiv.org/PS_cache/math/pdf/0111/0111139v1.pdf) 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?



Module categories over $Rep(G)$.
Sebastian: your definition of Rep^1(\tilde H) is absolutely correct. If you have
a representation of G you can restrict it to H and consider it as a representation
of \tilde H (this operation is called inflation). Now you can tensor it with any
representation of \tilde H; this tensoring preserves Rep^1(\tilde H); this is 
a module category structure (same thing was explained above by t3suji).


The category Rep(H) considered as a module category over Rep(G) is indecomposable.
It corresponds to subgroup H and trivial cocycle \omega (so \tilde H is a direct product
of H and multiplicative group G\_m).


