Let $G$ be a group and let $E$ be a homogeneous principal $G$-set. Fix $a\in E$. For $\gamma\in\text{Aut}(G)$, let $s_a(\gamma)$ be the permutation of $E$ defined by $$s_a(\gamma)(ga)=\gamma(g).a$$ for all $g\in G$.
I don't understand the definition of $s_a(\gamma)$: the way it's worded makes it seem like its domain is $G$. What is the value $s_a(\gamma)(x)$ for a generic element $x\in E$? Is there a cleaner way of defining $s_a$?
If a principal homoegeneous $G$-set is what I think it is, here's the explanation: since $E$ is a principal homogeneous $G$-set, once $a \in E$ has been chosen, every element of $E$ is of the form $ga$ for some $g \in G$. So given $\gamma \in {\rm Aut}(G)$, we would have $s_a(\gamma)\colon E \to E$. If you don't want to write $s_a(\gamma)(ga) = \gamma(g)\cdot a$, you could look at the orbit map of $a$, $\phi_a\colon G \to E$ (given by $\phi_a(g) = g\cdot a$), which is a bijection, and write $$s_a(\gamma)(x) = \gamma(\phi_a^{-1}(x))\cdot a $$instead, where now $x \in E$.