Let $G$ be a finite group, and let $\sigma:G\to G$ be a homomorphism of groups. There is an induced map in cohomology $\sigma^*:H^*(G,\mathbb{F}_p)\to H^*(G,\mathbb{F}_p)$. We also have the Bockstein $\delta:H^n(G,\mathbb{F}_p)\to H^{n+1}(G,\mathbb{F}_p)$ which is the connecting homomorphism associated to the short exact sequence of $\mathbb{F}_p$-modules
$$0\to\mathbb{F}_p\to\mathbb{Z}/p^2\mathbb{Z}\to\mathbb{F}_p\to 0$$
Does $\sigma^*$ commute with $\delta$? That is, do we have $\sigma^*\circ\delta=\delta\circ\sigma^*$ as maps from $H^n(G,\mathbb{F}_p)\to H^{n+1}(G,\mathbb{F}_p)$?
It seems like these maps should commute, but my understanding of the Bockstein is a bit clouded for me to see this clearly.