$L[G]\otimes_{K[G]} M \cong L[G]\otimes_{K[G]} N \Rightarrow M \cong N$

Question

Below is the problem with which I am struggling.

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Let $G$ be a finite group.

Let $L/K$ be an extension of fields of characteristic $0.$

Let $M$ and $N$ be finitely generated $K[G]$-modules.

Claim. If $L\otimes_K M \cong L\otimes_K N$ as $L[G]$-modules, then $M\cong N$ as $K[G]$-modules.

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First, I tried to prove it in the case where $L/K$ is finite by considering $\phi\circ\theta: M\to N$ where

$\bullet\; \theta:M\to L\otimes_K N$ is given by $\theta(m)=\phi(1\otimes_K m)$ where $\phi$ is the given isomorphism, and

$\bullet\; \phi:L\otimes_K N\to N$ is determined by $\phi(l\otimes_K n) = Tr_{L/K}(l)\cdot n.$

But this didn't work out...

I am aware that there are a few similar posts, such as:

1. Can extending a finite ground field make modules isomorphic?

2. https://mathoverflow.net/questions/28469/hilbert-90-for-algebras

My problem with (1) is that, the OP gives a "very easy argument" for my claim but I don't understand what "determinant" means when the ground ring is non-commutative.

As for (2), it proves a much stronger result and makes use of many things I don't know about yet.

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I would be immensely grateful if someone could show me a proof in my particular setting!