I am trying to understand a computation of the paper Twists of Elliptic Curves by Kronberg, Soomro and Top.
Let $E,E'$ be a elliptic curves over $\mathbb{F}_3$ given by $$ E: \quad y^2 = x^3 - x $$ and $$ E': \quad y^2 = x^3 + x. $$
Further assume that $K = \mathbb{F}_q$ be an extension of $\mathbb{F}_3$ of odd degree.
We consider the automorphism $\Psi_{u,r}: E \to E$ given by $\Psi_{u,r}(x,y) = (u^2 x + r, u^3 y)$ (where $u^4=1$ and $r \in \mathbb{F}_3$).
In the proof of Proposition 2.1. in the paper, it is said that for $\psi: E \to E'$, $(x,y) \mapsto (ix,-iy)$ (with $i^2 = -1$), we have $\left( \operatorname{Fr}(\psi) \right)^{-1} \circ \psi = \Psi_{i,0}$ (where $Fr$ is the field automorphism raising any element to its qth power, according to the paper). Now I am trying to verify this equality.
Since $\psi(x,y) = (ix,-iy)$, I concluded $\operatorname{Fr}(\psi)(x,y) = (i^q x, (-i)^q y) = (-ix, iy)$ (since $K/\mathbb{F}_3$ is odd) and therefore $\left( \operatorname{Fr}(\psi) \right)^{-1}(x,y) = (ix,-iy)$. This would give $\left( \operatorname{Fr}(\psi) \right)^{-1} \circ \psi (x,y) = (-x,-y)$. However, it is $\Psi_{i,0}(x,y) = (-x,-iy)$, so I must have made a mistake somewhere.
Could you help me finding my error?