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 $\hat{\pi}$ be the Aubert involution of $\pi$. We compute explicitly the Langlands data of $\hat{\pi}$ and the $A$-parameter $\hat{\psi}$ of $\hat{\pi}$. We investigate whether $\psi$ and $\hat{\psi}$ are symmetric. Although symmetry holds for large classes of parameters, it does not hold in general.