Converse of Clifford's theorem for a semidirect product Suppose that a group $G$ is a semidirect product $G = N \rtimes H$ with $N \trianglelefteq G$.
Let $\mathbb{F}$ be a field.
Say $V$ is a finite-dimensional $\mathbb{F}[G]$-module such that $V \downarrow N$ and $V \downarrow H$ are completely reducible.
If $H$ is also normal in $G$ (so $G = N \times H$), then I know that $V$ is completely reducible.
When $H$ is not normal, is there an example where $V$ is not completely reducible? It seems like there should be. I would be especially interested in the case where $G$ is finite.
 A: Let $N$ be a group.
Let $V$ and $W$ be finite-dimensional $\mathbb{F}[N]$-modules such that $V$ and $V \otimes W$ are c.r., but $W$ is not.
Consider $V \otimes W$ as a $\mathbb{F}[N \times N]$-module. We have $G = N \times N = A \rtimes B$, where $A = N \times 1$ and $B$ is the diagonal subgroup $B =\{ (x,x) : x \in N \}$.
Then $(V \otimes W) \downarrow B$ and $(V \otimes W) \downarrow A$ are c.r., since $V \otimes W$ and $V$ are c.r. $\mathbb{F}[N]$-modules.
But $V \otimes W$ is not c.r., since the restriction to $1 \times N$ is a direct sum of copies of $W$.
Some examples of $N$, $V$, $W$ such that $V$ and $V \otimes W$ are c.r., $W$ is not c.r. can be found in the following paper of Serre:

J.-P. Serre, Semisimplicity and tensor products of group representations: converse theorems. With an appendix by Walter Feit. J. Algebra 194 (1997), no. 2, 496–520.

Serre also proves that for such $V$ and $W$ we must have $\dim W \equiv 0 \mod{p}$.
Here is another example:

Example: Let $N = SL(2,\mathbb{F})$, where $\mathbb{F}$ is a field of characteristic two with $|\mathbb{F}| > 2$. Let $V$ be the $2$-dimensional natural module and $W = S^2(V)$. Then $V$ is irreducible, and $W$ is a nonsplit extension $$0 \rightarrow V^{[2]} \rightarrow S^2(V) \rightarrow \mathbb{F} \rightarrow 0.$$ A calculation shows that $V \otimes W \cong V \oplus (V \otimes V^{[2]})$ is completely reducible. (Here $V \otimes V^{[2]}$ is irreducible by Steinberg's tensor product theorem, or by direct calculation.)

