\Gamma_k^\alpha$ for $i=1,2$. Then, $\mathscr{C}_1=\mathscr{C}_2$ if and only if there exists a matrix $S\in GL(k,q)$ such that $\tau_1=S\cdot\tau_2\cdot S^{-1}$ and $P_1S=P_2$.

“$\Rightarrow$” Since $\mathscr{C}_1=\mathscr{C}_2$, there exists a matrix $S\in GL(k,q)$ such that $$\begin{aligned}
 S_t [(P_1)_t,(P_1\tau_1)_t,\dots,(P_1\tau_1^{n-1})_t]&=
[(P_1S)_t,(P_1\tau_1S)_t,\dots,(P_1\tau_1^{n-1}S)_t]\\
    &=[(P_2)_t,(P_2\tau_2)_t,(P_2\tau_2^2)_t,\dots,(P_2\tau_2^{n-1})_t]
   \end{aligned}$$

From the first columns we deduce that $P_1S=P_2$, that is, $P_2S^{-1}=P_1$. Furthermore, $P_2\tau_2^i=P_1\tau_1^iS=\left(P_2S^{-1}\right)\tau_1^iS
=P_2\left(S^{-1}\tau_1S\right)^i$, with $i=1,\dots,n-1$.

Thus $\{ P_2,P_2\tau_2,\dots,P_2\tau_2^{k-1}\}$ are linearly independent vectors of $\F^n_q$ and a vector $\vec{v}\in\F_q^n$ can be written as $\vec{v}=\sum_{i=0}^{k-1}\lambda_{ij}\left(P_2\tau_2^i\right)$. So we have [$$\begin{aligned}
\vec{v}\tau_2&=\sum\limits_{i=0}^{k-1}\lambda_{ij}\left(P_2\tau_2^i\right) \tau_2  &  &,& \vec{v}\left(S^{-1}\tau_1S\right)&=\sum\limits_{i=0}^{k-1}\lambda_{ij}\left(P_2\tau_2^i\right)\left(S^{-1}\tau_1S\right) \\
&=\sum\limits_{i=0}^{k-1}\lambda_{ij}\left(P_2\tau_2^{i+1}\right) &  &&
& =\sum\limits_{i=0}^{k-1}\lambda_{ij}\left(P_2\left(S^{-1}\tau_1S\right)^i\right)\left(S^{-1}\tau_1S\right)  \\
&=\sum\limits_{i=0}^{k-1}\lambda_{ij}\left(P_2\left(S^{-1}\tau_1S\right)^{i+1}\right) &  & &
& =\sum\limits_{i=0}^{k-1}\lambda_{ij}\left(P_2\left(S^{-1}\tau_1S\right)^{i+1}\right)\end{aligned}$$]{}

Since $\vec{e_j}\tau_2=\vec{e_j}S^{-1}\tau_1S$ for any canonical vector $\vec{e_j}$ with $j=1,\dots,k$, we conclude that $S^{-1}\tau_1S=\tau_2$.\
“$\Leftarrow$” Since there is a matrix $S\in GL(k,q)$ such that $\tau_2=S^{-1}\cdot\tau_2\cdot S$ y $P_1=P_2S^{-1}$, we see that $$\begin{aligned}
[(P_2)_t,(P_2\tau_2)_t,\dots,(P_2\tau_2^{n-1})_t]&=[(P_1S)_t,(P_1\tau_1S)_t,(P_1\tau_1^2S)_t,\dots,(P_1\tau_1^{n-1}S)_t]\\
&=S_t [(P_1)_t,(P_1\tau_1)_t,(P_1\tau_1^2)_t,\dots,(P_1\tau_1^{n-1})_t].
   \end{aligned}$$ By Theorem \[GMaruta:2\], the matrices defined by $(\tau_1,P_1,n)$ and $(\tau_2,P_2,n)$ are parity check matrices of skew $\alpha$-cyclic codes which differ by an invertible matrix, i.e. they correspond to the same code. Hence $\mathscr{C}_1=\mathscr{C}_2$.

Unlike the commutative case (see [@M1 Theorem 4]), when $\theta\not= id$ we have to consider the minimal polynomial $m_\tau$ of a semi-linear map $\tau$ (see [@TT2 Proposition 3.2]) instead of the characteristic polynomial. On the other hand, a 