Carmichael numbers for $\mathrm{G}\mathrm{L}(m)$

We propose a generalization of Carmichael numbers, where the multiplicative group ${\mathbb{G}}_{\mathrm{m}}=\mathrm{G}\mathrm{L}(1)$ is replaced by $\mathrm{G}\mathrm{L}(m)$ for $m\ge 2$. We prove basic properties of these families of numbers and give some examples.