Lifting subgroups of symplectic groups over ℤ/ℓℤ

For a positive integer g, let Sp_(2g)(R) denote the group of 2g×2g symplectic matrices over a ring R. Assume g≥2. For a prime number ℓ, we give a self-contained proof that any closed subgroup of Sp_(2g)(ℤℓ) which surjects onto Sp2g(ℤ/ℓℤ) must in fact equal all of Sp_(2g)( ℤℓ). The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties