Artin's Conjecture for Abelian Varieties with Frobenius Condition

$A$ be an abelian variety over a number field $K$ of dimension $r$, $a_1,\dots ,a_g\in A(K)$ and $F\mathrm{/}K$ a finite Galois extension. We consider the density of primes $\mathfrak{p}$ of $K$ such that the quotient $\bar{A}(k(\mathfrak{p}))\mathrm{/}⟨{\bar{a}}_1,\dots ,{\bar{a}}_g⟩$ has at most $2r-1$ cyclic components and $\mathfrak{p}$ satisfies a Frobenius condition with respect to $F\mathrm{/}K$, where $\bar{A}$ is the reduction of $A$ modulo $\mathfrak{p}$, $k(\mathfrak{p})$ is the residue class field of $\mathfrak{p}$ and $⟨{\bar{a}}_1,\dots ,{\bar{a}}_g⟩$ is the subgroup generated by the reductions ${\bar{a}}_1,\dots ,{\bar{a}}_g$. We develop a general framework to prove the existence of the density under the Generalized Riemann Hypothesis.