We are given \(P(A) = \frac{1}{7}\), \(P(B) = \frac{5}{7}\), and \(P(A \cap B) = \frac{4}{7}\). We need to find \(P(\overline{A}|B)\).

By definition of conditional probability, \(P(\overline{A}|B) = \frac{P(\overline{A} \cap B)}{P(B)}\).

We know that \(P(B) = P(A \cap B) + P(\overline{A} \cap B)\). Therefore, \(P(\overline{A} \cap B) = P(B) - P(A \cap B)\).

Substituting the given values, we have \(P(\overline{A} \cap B) = \frac{5}{7} - \frac{4}{7} = \frac{1}{7}\).

Now, we can find \(P(\overline{A}|B) = \frac{P(\overline{A} \cap B)}{P(B)} = \frac{\frac{1}{7}}{\frac{5}{7}} = \frac{1}{5}\).