Comparison of component groups of $\mathrm{\ell }$-adic and mod $\mathrm{\ell }$ monodromy groups

Let $\{{\rho }_{\mathrm{\ell }}:{\mathrm{G}\mathrm{a}\mathrm{l}}_K\to {\mathrm{G}\mathrm{L}}_n({\mathbb{Q}}_{\mathrm{\ell }}){\}}_{\mathrm{\ell }}$ be a semisimple compatible system of $\mathrm{\ell }$-adic representations of a number field $K$ that is arising from geometry. Let ${\mathbf{\text{G}}}_{\mathrm{\ell }}\subset {\mathrm{G}\mathrm{L}}_{n,{\mathbb{Q}}_{\mathrm{\ell }}}$ and $\widehat{\underset{‾}{G_{\mathrm{\ell }}}}\subset {\mathrm{G}\mathrm{L}}_{n,{\mathbb{F}}_{\mathrm{\ell }}}$ be respectively the algebraic monodromy group and full algebraic envelope of ${\rho }_{\mathrm{\ell }}$. We prove that there is a natural isomorphism between the component groups ${\pi }_0({\mathbf{\text{G}}}_{\mathrm{\ell }})\simeq {\pi }_0(\widehat{\underset{‾}{G_{\mathrm{\ell }}}})$ for all sufficiently large $\mathrm{\ell }$.