Formula for the Frobenius-Schur indicator of a finite group?

For one answer, here is a theorem due to Thompson and Willems (Bilinear forms in characteristic $p$ and the Frobenius-Schur indicator, Lecture Notes in Mathematics 1185, pg. 221-230).

For an irreducible self-dual $kG$-module $V$, set $\varepsilon(V) = 1$ if $G$ preserves a nonzero symmetric bilinear form on $V$, and $\varepsilon(V) = -1$ if $G$ preserves a nonzero alternating form on $V$.

Theorem: Let $V$ be an irreducible self-dual $kG$-module with Brauer character $\phi$. There exists an ordinary irreducible character $\chi$ such that

(1) $\chi$ is real valued.

(2) The decomposition number $d(\chi, \phi)$ is odd.

Furthermore, any ordinary irreducible character $\chi$ satisfying (1) and (2) has the property that $\nu_2(\chi) = \varepsilon(V)$.

So assuming you know the ordinary character table of $G$, and the Frobenius-Schur indicators and decomposition numbers of all irreducible characters; you can give an answer. I don't know if there are any other general results.

The answer to the question can be found in Section 14, Frobeius-Schur indicator for Brauer characters, starting on page 320, of the book "Group Representations, Volume 4" by Gregory Karpilovsky. I don't have easy access to this book myself (our library only has Volume 1, and that is in two parts), but fortunately the Google Books preview is letting me see all 14 pages of that section.

It says on page 321, "It is natural to ask whether there is a Frobenius-Schur indicator for the Brauer character $\beta$ of $G$ afforded by $V$, which allows us to determine whether $V$ is of symmetric of skew symmetric type. It turns out that the answer is positive and the required indicator may be taken to be the Frobenius-Schur indicator of a certain irreducible ${\mathbb C}$-character of $G$ closely related to $\beta$."

The theory of all this follows and takes about 12 pages, so it is not for the faint-hearted.

A Google search for "Frobenius Schur indicator Brauer character" comes up with several relevant pointers to technical papers on this topic, so it seems to be known to specialists.