Let $H$ and $K$ be affine group schemes over a field $k$ of caracteristic zero. Let $\varphi:H\to Aut(K)$ be a group action. Then we can form the semi-direct product $G = K\ltimes H$.

Problem: Describe the tannakian category $Rep_k(G)$ in terms of $Rep_k(K)$, $Rep_k(H)$ and $\varphi$.

If $\varphi$ is the trivial action, $G$ is the direct product $K\times H$. In this case, $Rep_k(G) = Rep_k(K) \boxtimes Rep_k(G)$ is just the Deligne tensor product of the tannakian categories.

Question: What happens in the opposite scenario where the action $\varphi$ is faithful? Can one characterize this situation in terms of Ext groups of simple objects in $Rep_k(K)$ and $Rep_k(H)$?

I'm mostly insterested in the case where $K$ and $H$ are extensions of $\mathbb{G}_m$ by a pro-unipotent group but I have no idea where to start.

Let $H$ and $K$ be affine group schemes over a field $k$ of characteristic zero. Let $\varphi:H\to Aut(K)$ be a group action. Then we can form the semi-direct product $G = K\ltimes H$.

Problem: Describe the tannakian category $Rep_k(G)$ in terms of $Rep_k(K)$, $Rep_k(H)$ and $\varphi$.

If $\varphi$ is the trivial action, $G$ is the direct product $K\times H$. In this case, $Rep_k(G) = Rep_k(K) \boxtimes Rep_k(G)$ is just the Deligne tensor product of the tannakian categories.

Question: What happens in the opposite scenario where the action $\varphi$ is faithful? Can one characterize this situation in terms of Ext groups of simple objects in $Rep_k(K)$ and $Rep_k(H)$?

I'm mostly insterested in the case where $K$ and $H$ are extensions of $\mathbb{G}_m$ by a pro-unipotent group but I have no idea where to start.