Let $k$ be a field and let $G$ be an algebraic group over $k$. 


I encountered the following notion in an article:


"Let $\psi: \text{Gal}(\overline{k}/k) \to \text{Int}(\overline{G})$ be a cocycle. The **twisted group** $\_\psi G$..."


How is this "twisted group" defined?



Definition of "twisted group"
You can find the definitions in Serre's book "Galois cohomology". The relevant sections are I.5.3 - I.5.7. The idea is that if you have a group $G$ acting on a group $A$ and a cocycle $\psi\in H^1(G,A)$. The twisted group $\_\psi A$ has the same group structure as $A$ but it has a twisted action by $G$, which is given by
$$g\*a = \psi(g) \cdot (g \cdot a) \cdot \psi(g)^{-1}.$$
There is also a previous question which asked if there is an analog of this for outer automorphisms: [Substitute for Serre’s twisting when the “twisting” is outer](https://mathoverflow.net/questions/10048/substitute-for-serres-twisting-when-the-twisting-is-outer)


