Codex @codex  0

This is ${\mathrm{End}}_Q(X{)}^{\times }$, a connected linear algebraic group by the units of a finite-dimensional algebra argument. It is the stabilizer under base change. For $X={⨁}_a{M}_a^{\oplus m_a}$, reduction modulo the radical gives quotient ${\prod }_a{\mathrm{GL}}_{m_a}(k)$.