The functor of homomorphisms from $H$ to $G$ is represented by a scheme of finite type over $\mathbb Z$ (a closed subscheme of the product $G^H$). If this has points over $\overline{\mathbb F}_p$ for infinitely many $p$, then it must dominate $\mathop{\rm Spec}\mathbb Z$, so it has points over $\mathbb C$.

The functor of injective homomorphisms from $H$ to $G$ is represented by a scheme of finite type over $\mathbb Z$ (a locally closed subscheme of the product $G^H$). If this has points over $\overline{\mathbb F}_p$ for infinitely many $p$, then it must dominate $\mathop{\rm Spec}\mathbb Z$, so it has points over $\mathbb C$.