Construction of Abelian varieties with given monodromy

Let $\rho$ be a finite-dimensional faithful representation of a semisimple algebraic group $G$. By means of a deformation argument, we show that there exists a family of Abelian varieties over a smooth and projective curve over the algebraic closure of a prime field of positive characteristic, such that its $\mathrm{\ell }$-adic monodromy group covers $G$ and its $\mathrm{\ell }$-adic monodromy representation contains $\rho$.