Symmetry of Arthur parameters under Aubert involution

We consider a nontempered $A$-parameter $\psi$ of $SO(2n+1,F)$ of a certain type and the base point representation $\pi$ in the $A$-packet of $\psi$. Let $\widehat{\pi }$ be the Aubert involution of $\pi$. We compute explicitly the Langlands data of $\widehat{\pi }$ and the $A$-parameter $\widehat{\psi }$ of $\widehat{\pi }$. We investigate whether $\psi$ and $\widehat{\psi }$ are symmetric. Although symmetry holds for large classes of parameters, it does not hold in general.