Step 1 :Given that hexagonal prism B is the image of hexagonal prism A after dilation by a scale factor of \(\frac{1}{4}\), and the volume of hexagonal prism B is \(2 \mathrm{~m}^{3}\).
Step 2 :The ratio of the volumes of the two prisms is the cube of the scale factor, which is \(\left(\frac{1}{4}\right)^3 = \frac{1}{64}\).
Step 3 :Let the volume of hexagonal prism A be \(V_A\). Then, \(V_A \times \frac{1}{64} = 2\).
Step 4 :Solving for \(V_A\), we get \(V_A = 2 \times 64 = 128\).
Step 5 :\(\boxed{128 \mathrm{~m}^{3}}\) is the volume of hexagonal prism A.