Definition: Let $A=A=\bigoplus_{j=0}^{\infty} A_j$ be a connected $\mathbb{N}$-graded $k$-graded algebra and let $\phi\in\text{Aut($A$)}$ be a graded automorphism of degree zero. A new graded algebra $B$ is formed as follows:
as vector spaces, $B=A$, y
we define a multiplication $\ast$ on $B$ by $a_1\ast a_2 \ast\cdots\ast a_m=a_1\phi(a_2)\phi^2(a_3)\cdots\phi^{m-1}(a_m),$
for all $a_1,\ldots,a_m\in A_1, m\in\mathbb{N},$ and distributivity to extend $\ast$ to the rest of $B$.
We call $B$ the twist of $A$ by the automorphism $\phi.$
We note that twisting a graded algebra by an automorphis is reflexive and symmetric, but usually it is not transitive.
What I understand by transitivity is that if I have a graded algebra $A$ I twist it by an automorphism $\phi$ I get a twisted algebra $B$ then I twist the graded algebra $B$ by another automorphism $\psi$ and get a twisted graded algebra $C.$ So, it is not possible to find an automorphism such that twisting the graded algebra $A$ corresponds to the graded algebra $C.$
I am trying to construct or find a counterexample but it has been impossible. Any suggestions or help? Thank you very much in advance.