Linear immanant converters on skew-symmetric matrices of order $4$

Let $Q_n$ denote the space of all $n\times n$ skew-symmetric matrices over the complex field $\mathbb{C}$. It is proved that for $n = 4$, there are no linear maps $ T :Q_4\to Q_4$ satisfying the condition $ d_{\chi'} ( T (A) ) =d_{\chi} (A) $ for all matrices $ A\in Q_4$, where $\chi, \chi' \in \{1, \epsilon, [2,2]\}$ are two distinct irreducible characters of $S_4$. In the case $\chi=\chi'=1$, a complete characterization of the linear maps $T :Q_4\to Q_4$ preserving the permanent is obtained. This case is the only one corresponding to equal characters and remaining uninvestigated so far.