let $M_{2}(\mathbb{F}_{p})$ be the vector space of 2$\times$2 matrices over the finite field $\mathbb{F}_{p}$ where $p$ is a prime number, and let $GL_{2}(\mathbb{F}_{p})$ be the group of invertible 2$\times$2 matrices over $\mathbb{F}_{p}$. We let $GL_{2}(\mathbb{F}_{p})$ act on $M_{2}(\mathbb{F}_{p})$ by conjugation. My question is how to compute the cohomology group $H^{1}(GL_{2}(\mathbb{F}_{p}), M_{2}(\mathbb{F}_{p}))$ ?