The correct answer is A) Immediately after the towing vessel sounds her fog signal. According to the International Regulations for Preventing Collisions at Sea (COLREGS) and the Inland Navigation Rules, a manned vessel being towed is required to sound her fog signal immediately after the towing vessel sounds her own fog signal. This ensures that the fog signals of both vessels are properly coordinated and helps other vessels in the vicinity identify the presence and relative positions of the towing and towed vessels. The other options are incorrect because B) immediately before the towing vessel is out of sequence, C) as close to the mid-cycle is less precise than the required immediate timing, and D) any time is too broad and does not follow the specific regulation.