Let $G$ be a finite group and $\pi$ an irreducible complex representation. The Frobenius-Schur indicator of $\pi$ is defined as:

$$ \nu_2(\pi):=\frac{1}{|G|} \sum_{g \in G} \chi_{\pi}(g^2) $$
with $\chi_{\pi}$ the character of $\pi$.

Note that the map $s: g \mapsto g^2$ is well-defined on the conjugacy classes as $\tilde{s}: C(g) \mapsto C(g^2)$ because $(hgh^{-1})^2 = hg^2h^{-1}$. Let $\chi_1, \cdots, \chi_r$ be the irreducible characters of $G$ (with $\chi_i = \chi_{\pi_i}$), and $C_1, \cdots, C_r$ be the conjugacy classes, with $\chi_1$ the trivial and $C_1 = C(1)$. The character table of $G$ is given by the matrix $(\chi_{i,j})$ with $\{ \chi_{i,j} \} = \chi_i(C_j)$. The map $\tilde{s}$ induces a map $m$ on $\{1,2, \cdots, r \}$ such that $\tilde{s}(C_j) = C_{m(j)}$. It follows that the Frobenius-Schur indicator $\nu_2$ is completely determined by the character table $(\chi_{i,j})$ and the map $m$ as follows:

$$ \nu_2(\pi_i):=\frac{1}{|G|} \sum_{j} |C_{j}|\chi_{i,m(j)} = \sum_j \frac{\chi_{i,m(j)}}{\sum_i |\chi_{i,j}|^2}$$ because $|C_j| = |G|/\sum_i |\chi_{i,j}|^2$.

Note that the character table alone is not sufficient to determine $\nu_2$. For example, the quaternion group $Q_8$ and the dihedral group $D_4$ have same character table, but the first admits an irreducible complex representation with Frobenius-Schur indicator $-1$ (in fact it is the smallest such finite group) whereas the second not. But these do not have the same class type $(1,2,4A,4B,4)$ for the first and $(1,2A,2B,2C,4)$ for the second (a class is of type $nX$ if its elements has order $n$).

**Question**: Is the Frobenius-Schur indicator $\nu_2$ completely determined by the character table *including* the class types? If so, what is the formula?

It is "suggested" true by the section 71.12-5 in GAP manual, as GAP seems to need these data only to compute $\nu_2$.

`Display(CharacterTable("A5"));;`

you see the character table of $A_5$ along with the power maps. Perhaps with a computer search you could find two groups with same character tables and class types, but different Frobenius-Schur indicators. $\endgroup$