I was searching through the small groups database in GAP to find counterexamples to a certain conjecture (which is not important here). I was checking non-nilpotent solvable groups and noticed that all the permutation modules ($1_H^G$ for $H\leq G$) in characteristic 2 have self-dual socles. Obviously this is true in characteristic 0 where the modules are completely reducible (since $1_H^G$ is self-dual). However, this is not the case in characteristic 3 with $GL(2,3)$ being the smallest example. I also found counterexamples in characteristic 5 (the smallest of order 320), and for non-solvable groups starting with $PSL(3,2)$. However, I didn't find any counterexamples for solvable groups in characteristic 2 (up to $|G|=700$) which I found to be quite odd.

It seems like this should have been conjectured or proven somewhere (if it's true), but I was unable to find any references to it in the literature—likely because I don't know what to search for. So my questions are

Is there a counterexample?

If not, has this been investigated in the literature?