Let $G$ be a connected, simply-connected simple group over $\mathbb{C}$. The structure/classification of unipotent orbits of $G$ is described in many places.

Now, I'd like to know the structure/classification of unipotent orbits of $Aut(G)$, in particular those which is not in the component connected to the identity. Where can I find the descriptions?

Even more concretely, I am the most interested in the case $G=SL(2N+1)$ (as the corresponding twisted affine algebra $A_{2n}^{(2)}$ behaves rather exceptionally.)

Concerning this case, I have a following guess: Let $\sigma_B$, $\sigma_C$ elements in $Aut(SL(2N+1))$ disconnected to the identity, such that $SL(2N+1)^B=SO(2N+1)$ and $SL(2N+1)^C=Sp(2N)$, respectively.

There is a standard order-preserving map between the sets of special unipotent orbits of $B_N$ and $C_N$. Take $O_B$ and $O_C$ special unipotent orbits of $B_N$ and $C_N$ related this way, an pick an element from each, $x_B$ and $x_C$. They determine naturally elements $\xi_B$, $\xi_C$ of $Aut(G)$ via

$\xi_{B,C}(.)=x_{B,C} \sigma_{B,C}(.) x_{B,C}{}^{-1}$,

Then, I guess $\xi_B$ and $\xi_C$ are conjugate as elements of $Aut(G)$, which would give a "geometric" description of the map between the sets of special unipotent orbits of $B_N$ and $C_N$ ... Is this a known theorem?

Let $G$ be a connected, simply-connected simple group over $\mathbb{C}$. The structure/classification of conjugacy classes of $G$ is described in many places.

Now, I'd like to know the structure/classification of conjugacy classes of $Aut(G)$, in particular those which is not in the component connected to the identity. Where can I find the descriptions?

Even more concretely, I am the most interested in the case $G=SL(2N+1)$ (as the corresponding twisted affine algebra $A_{2n}^{(2)}$ behaves rather exceptionally.)

Concerning this case, I have a following guess: Let $\sigma_B$, $\sigma_C$ elements in $Aut(SL(2N+1))$ disconnected to the identity, such that $SL(2N+1)^B=SO(2N+1)$ and $SL(2N+1)^C=Sp(2N)$, respectively.

There is a standard order-preserving map between the sets of special unipotent orbits of $B_N$ and $C_N$. Take $O_B$ and $O_C$ special unipotent orbits of $B_N$ and $C_N$ related this way, an pick an element from each, $x_B$ and $x_C$. They determine naturally elements $\xi_B$, $\xi_C$ of $Aut(G)$ via

$\xi_{B,C}(.)=x_{B,C} \sigma_{B,C}(.) x_{B,C}{}^{-1}$,

Then, I guess $\xi_B$ and $\xi_C$ are conjugate as elements of $Aut(G)$, which would give a "geometric" description of the map between the sets of special unipotent orbits of $B_N$ and $C_N$ ... Is this a known theorem?