The outer automorphism group $\textrm{Out}(G)$ of a finite group $G$ act on the character table by permuting columns (conjugacy classes) and rows (representations). As a result, there is an homomorphism $\psi:\textrm{Out}(G)\rightarrow\textrm{CTS}(G)$, where $\textrm{CTS}(G)$ is the symmetry group of the character table of $G$. When is this homomorphism $\psi$ surjective or injective?
More specifically, given $\sigma\in\textrm{CTS}(G)$, are there conditions involving properties of the conjugacy classes and representations that are being permuted by $\sigma$, that allows one to know that exists $\alpha\in\textrm{Out}(G)$ such that $\psi\cdot\alpha=\sigma$? For example, one can show that the centralizer of the conjugacy classes that are permuted by an outer automorphism are isomorphic, is the converse true? This is consistent with some few cases that I considered, in particular with $Q_8$ and $D_4$, that share the same character table but have $\textrm{Out}(Q_8)=S_3$ and $\textrm{Out}(D_4)=\mathbb{Z}_2$.