Step 1: Understanding volume strength of hydrogen peroxide.
The volume strength of hydrogen peroxide refers to the volume of oxygen gas that 1 liter of the solution can produce when decomposed. A 20-volume solution of H$_2$O$_2$ means that 1 liter of the solution will produce 20 liters of oxygen gas. From the molar mass of hydrogen peroxide (34 g/mol), the strength can be calculated. The correct strength of a 20-volume solution is 60.71 g/L.
Step 2: Analyzing the options.
(A) 60.71 g/L: This is the correct answer, as the volume strength and mass concentration are related as described.
(B) 6.8 g/L: This is incorrect, as it is much too low for a 20-volume solution.
(C) 68 g/L: This is also incorrect, as it is too high for the given solution strength.
(D) 6.071 g/L: This is incorrect, as it does not match the correct concentration for a 20-volume solution.
Step 3: Conclusion.
The correct answer is (A) 60.71 g/L.