What can be said about Schur indices, given only the character table? Let $\chi$ be an irreducible (complex) character of a finite group, $G$. The Schur index $m_{K}(\chi)$ of $\chi$ over the field $K$ is the smallest positive integer $m$ such that $m\chi$ is afforded by a representation over the field $K(\chi)$. The most interesting case is $K=\mathbb{Q}$. Given the character table, or only the particular character one is interested in, one can usually derive bounds for $m(\chi)=m_{\mathbb{Q}}(\chi)$. For example, $m(\chi)$ divides $\chi(1)$ and $n[\chi^n,1_G]$ for all $n\in \mathbb{N}$ (Fein), and the Benard-Schacher Theorem tells us that $\mathbb{Q}(\chi)$ contains a primitive $m(\chi)$-th root of unity.
On the other hand, the example of the quaternion group $Q_8$ and the dihedral group $D_8$ shows that two groups might have identical character tables, but corresponding characters with different Schur indices.  I am curious wether there are examples that are even worse than this.
Notation: To state this more precisely, I'll make the following assumptions: We are given two finite groups $G$ and $H$, such that there is a bijection $\tau\colon {\rm Cl}(G) \to {\rm Cl}(H)$ from the classes of $G$ to the classes of $H$, and such that $\psi \mapsto \psi \circ \tau$ is a bijection ${\rm Irr}(H)\to {\rm Irr}(G)$. Now:

Is there an example with $m(\chi)/m(\chi\circ\tau)\notin \{1,2,1/2\}$ for some $\chi\in {\rm Irr}(H)$?
Is there an example with $G$ of odd order and $m(\chi) / m(\chi\circ\tau)\neq 1$ for some $\chi \in {\rm Irr}(H)$?

Now let us assume that we know the power maps of the character table. These are the maps $\pi_n^G\colon {\rm Cl}(G)\to {\rm Cl}(G)$ induced by $g\mapsto g^n$. (These maps are stored in the tables of the character table library of GAP.) Given these maps, one can compute $[\chi_C, 1_C]$ for cyclic subgroups $C\leq G$, for example. Also we can compute the Frobenius-Schur Indicator and thus the Schur index over $\mathbb{R}$.
Now assume that $\tau\circ \pi_n^G = \pi_n^H\circ \tau$ in the above situation (then $(G,H)$ is called a Brauer pair).

Is there a Brauer pair $(G,H)$ such that $m(\chi)/m(\chi\circ\tau)\neq 1$ for some $\chi\in {\rm Irr}(H)$?

I would appreciate any examples or (pointers to) results that show the impossibility of such examples.
Thanks
 A: The following is a theorem of K. Kronstein:

Theorem: for $k$ a number field or a nonarchimedean completion of a number field, if it is possible to detect the Schur index $m_k$ of all finite groups from their character table and power maps, then $m_k(\chi) \leq 2$ for all characters $\chi$ of finite groups.

In particular, for a finite group $G$, the map $\tau: G \times G \to G \times G$ defined by $\tau(x,y) = (x,y^{-1})$ induces a bijection on conjugacy classes preserving the power maps $x \mapsto x^n$ and a bijection on characters, but does not necessarily preserve Schur indices larger than 2. Thus, the map $\tau$ provides a positive answer to the first and third questions. It also provides a positive answer to the second question once one produces an example of a group of odd order with a character with Schur index greater than 2.
I provide Kronstein's proof here:
Suppose $\chi$ is a character of $G$, $k$ is a number field or nonarchimedean completion of a number field, and the Schur index $m = m_k(\chi)$ is at least $3$.
Let $K = k(\chi)$, let $V$ be an irreducible $KG$-module affording the character $m\chi$, and let $D = End_{KG}(V)$. It is a division algebra with order $m$ in the Brauer group of $K$.
Consider the characters $\chi \boxtimes \chi$ and $\chi \boxtimes \chi^\vee$ on $G \times G$. Then
$$End_{K(G\times G)}(V\boxtimes V) = D \otimes_K D.$$
Since $m>2$, $D \otimes_K D$ is not split, so $\chi\boxtimes \chi$ has Schur index greater than 1. However, $$End_{K(G \times G)}(V \boxtimes V^\vee) = D \otimes_K D^{op},$$
which splits over $K$, and thus $\chi \boxtimes \chi^\vee$ has Schur index 1.

Reference:
Karl Kronstein. Character tables and the Schur index. In Representation Theory of Finite Groups and Related Topics,
volume 21 of Proceedings of Symposia in Pure Mathematics, pages 97–98. American Mathematical Soc., 1971
