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 $F/K$ a finite Galois extension. We consider the density of primes $\frak p$ of $K$ such that the quotient $\bar{A}(k({\frak p}))/\langle \bar{a}_1,\dots,\bar{a}_g\rangle$ has at most $2r-1$ cyclic components and $\frak p$ satisfies a Frobenius condition with respect to $F/K$, where $\bar{A}$ is the reduction of $A$ modulo $\frak p$, $k(\frak p)$ is the residue class field of $\frak p$ and $\langle \bar{a}_1,\dots,\bar{a}_g\rangle$ 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.