Let $G$ be a finite group, $K$ a characteristic-$p$ algebraically closed field (say $p$ divides $|G|$), and let $M$ be a finite-dimensional $KG$-module.
What hypotheses are needed on $G$, $M$ to ensure that $H^{1}(G,M) \cong H^{1}(G,M^{*})$ where $M^{*}$ denotes the dual module Hom$(M,K)$?
In fact, in the situation I'm faced with, $G$ is a finite simple group and calculation suggests the stronger result that $Ext^1(M,N) \cong Ext^1(N,M)$ when $M$ and $N$ are irreducible, and at least one of $M$,$N$ is self-dual. There are counterexamples when the latter condition fails, e.g. Alt$_9$ has non-self-dual modules of dimension 8 and 20 with $Ext^1(M,N) \neq Ext^1(N,M)$.
I feel like this result should be well-known but I've been unable to find a reference for it yet.
A weaker result that would also satisfy me is any criterion to ensure $H^1(G,M) \neq 0 \Rightarrow H^1(G,M^{*}) \neq 0$.
Thanks in advance!