Let $K$ be a field (of course, of positive characteristic, unless you want a trivial question). Let $G$ be a finite group, and $V$ and $W$ be two completely reducible (finite-dimensional) representations of $G$ over $K$. Is the (interior) tensor product $V\otimes_K W$ a completely reducible representation of $G$ ?

I know that this holds for exterior tensor products: Let $K$ be a field. Let $G$ and $H$ be two finite groups, and $V$ and $W$ be two completely reducible (finite-dimensional) representations of $G$ and $H$, respectively, over $K$. Then, the tensor product $V\otimes_K W$ is a completely reducible representation of $G\times H$. (Proof: Combine Curtis/Reiner "Methods of Representation Theory I" Theorems 7.10 and 10.38 (i).)

Note that my above question is equivalent to the Jacobson radical of the group ring $KG$ being a coideal (the coalgebra structure on $KG$ is the canonical one, of course: $\Delta g=g\otimes g$). It may be total nonsense but unfortunately I don't have any nontrivial examples of modular representations to check with.

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