This is a question I received today by email, which somebody more experienced with finite group representations can probably answer directly. Take $F:=\mathbb{F}_q$ for some prime power $q$, so $G:=\mathrm{GL}(V)$ acts naturally on both $V:=F^n$ and its symmetric square in $V \otimes V$.

If $H$ is an absolutely irreducible subgroup of $G$, does it act absolutely irreducibly on the symmetric square of $V$ (and if so, what is a reference for this)?

Maybe it's enough here just to start with any absolutely irreducible representation of a finite group $H$ in $G$, not necessarily faithful? Also, I'm unclear about what happens in characteristic 2. Otherwise the statement looks reasonable but I can't readily find a reference in standard textbooks.