Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $p \neq 2$.

Let $V$ be a finite-dimensional irreducible $kG$-module. If $V \cong V^*$, then $V$ admits a nonzero $G$-invariant bilinear form $(-,-)$, unique up to scalar, such that $(-,-)$ is alternating or symmetric. Is there a formula or a general method to determine whether $(-,-)$ is going to be alternating or symmetric?

If $k = \mathbb{C}$, then the answer is yes: for a $\mathbb{C}G$-module $V$ with irreducible character $\chi$ we have the Frobenius-Schur indicator $$\nu_2(\chi) = \frac{1}{|G|} \sum_{g \in G} \chi(g^2)$$

which satisfies $\nu_2(\chi) \in \{-1, 0, 1\}$ and

- $\nu_2(\chi) \neq 0$ iff $V \cong V^*$.
- $\nu_2(\chi) = 1$ if there is a nonzero $G$-invariant orthogonal bilinear form on $V$.
- $\nu_2(\chi) = -1$ if there is a nonzero $G$-invariant alternating bilinear form on $V$.

I think similar things should be true when $p \not \mid |G|$. What about in general? Is this an open problem?