Let $p$ be a prime and $G=\mathrm{GL}_2(\mathbb{F}_p)$. Let $V$ be the $G$-module $\mathbb{F}_p^2$ with the standard action given by matrix multiplication. It seems that quite often the groups $H^i(H,V)$ vanish for $i=1,2$, but not always. For example, when $p=3$, the subgroup $H=C_3$ gives cohomology groups with $\mathbb{F}_3$ dimension 1 for $i=1,2$.
Having done a few Magma calculations, it seems like this only happens for subgroups $H$ whose order is 0 $\mod{p}$, but this doesn't fully characterise which subgroups have non-vanishing cohomology (the largest subgroup of $\mathrm{GL}_2(\mathbb{F}_3)$ satisfying $H^1(H,V) = H^2(H,V) = 0$ is $H=S_3$, but there are lots more subgroups with order divisible by 3).
I would like to know when these two cohomology groups vanish.
For practical purposes I can always just compute the cohomology and see if it vanishes, but I would still like to know if there is a criterion known that characterises these subgroups exactly.