Let $\alpha $ be a complex valued $2$-cocycle of finite order of a finite group $G.$ The nth Frobenius–Schur indicator of an irreducible $\alpha $-character of G is defined and its properties are investigated. The indicator is interpreted in general for $n =2$ and it is shown that it can be used to determine whether an irreducible $\alpha $-character is real-valued under the assumption that the order of $\alpha $ and its cohomology class are both $2$. A formula, involving the real $\alpha $-regular conjugacy classes of $G,$ is found to count the number of real-valued irreducible $\alpha $-characters of G under the additional assumption that these characters are class functions.

We study the numbers of involutions and their relation to Frobenius–Schur indicators in the groups $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius–Schur indicators are 1), and we observe this holds for all finite simple groups $G$ other than the groups $\unicode[STIX]{x1D6FA}^{\pm }(4m,q)$ with $q$ even. We prove computationally that for small $m$ this statement indeed holds for these groups by equating their character degree sums with the number of involutions. We also prove a result on a certain twisted indicator for the groups $\text{SO}^{\pm }(4m+2,q)$ with $q$ odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating functions and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$ for all $q$, and we use these to compute the asymptotic behavior for the number of involutions in these groups when $q$ is fixed and $n$ grows.