Electrostatic induction - how to quantify it with a theoretic integral expression or like?

Assume that a certain distribution of charges $rho$ generates an electrical field $E_{ext}$ in the surrounding space. We also note the corresponding generated potential $V_{ext}$.

Assume furthermore that a conductor A, with a definite shape and volume, is placed in field $E_{ext}$, and is connected to an infinitely far ground $G$ with a very thin electrical wire. The potential of G is supposed to be $0$.

By electrostatic induction, free charges in A move to G, and conductor A, after equilibrium, experiences a distribution of charge $sigma$ on its surface.

Assume furthermore that the charge $sigma$ does not influence $rho$, so that
$E_{ext}$ and $V_{ext}$ are given and fixed in this problem.

Question: how to characterize the total charge $Q = int_{S(A)} sigma dS$ of A, as a function of $E_{ext}$ or $V_{ext}$, using, e.g. surface integrals on A etc. ?