A supercharacter theory for involutive algebra groups

If $\mathscr{J}$ is a finite-dimensional nilpotent algebra over a finite field $\Bbbk$, the algebra group $P = 1+\mathscr{J}$ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If $\mathscr{J}$ is endowed with an involution $\widehat{\varsigma}$, then $\widehat{\varsigma}$ naturally defines a group automorphism of $P = 1+\mathscr{J}$, and we may consider the fixed point subgroup $C_{P}(\widehat{\varsigma}) = \{x\in P : \widehat{\varsigma}(x) = x^{-1}\}$. Assuming that $\Bbbk$ has odd characteristic $p$, we use the standard supercharacter theory for $P$ to construct a supercharacter theory for $C_{P}(\widehat{\varsigma})$. In particular, we obtain a supercharacter theory for the Sylow $p$-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by Andr\'e and Neto for the special case of the symplectic and orthogonal groups.