Is it possible to find an example of the following type?
Let $p$ be a prime and let $M_1$ and $M_2$ be elementary abelian finite $p$-groups (i.e., the additive group of two vector spaces over the field of characteristic $p$) of order $p^n$ and $p^m$, say.
Is it possible to embed $S=Sym(3)$ in $GL(n, p)$ and $GL(m, p)$ in such a way that in the corresponding action on $M=M_1\times M_2$ the only non-trivial $Alt(3)$-invariant subgroups are $M_1$ and $M_2$?
I asked about $Sym(3)$ because I thought it is the smallest possible order, but I would be happy even if it is possible to replace $S$ with any group of the form $H=\langle a\rangle\ltimes\langle b\rangle$, where $a$ and $b$ have prime power orders (and $H$ is not abelian)
I tried to find such an action with gap but it seems the prime $p$ and/or $n,m$ must be too large.
Do you know any example of this type?
Yes. Let $p$ be any prime with $3 \not\vert p-1$, such as $p=2$, let $n=2$ and $m=1$, with $M_1$ a nontrivial irreducible $S$-module, and $M_2$ the trivial module. (Or, for odd $p$, you could take $M_2$ to be the module in which $(1,2,3)$ acts trivially but $(1,2)$ does not.) These are the only examples for $S_3$.