1. **State the problem:** Simplify the expression $$\left( \frac{x^{5}y^{-3}}{2xy^{5}} \right)^{-4} \div \frac{4x^{6}y^{-10}}{\left( 3x^{-2}y^{2} \right)^{-3}}.$$\n\n2. **Simplify inside the parentheses:**\n\nFirst, simplify $$\frac{x^{5}y^{-3}}{2xy^{5}} = \frac{x^{5}}{x} \cdot \frac{y^{-3}}{y^{5}} \cdot \frac{1}{2} = \frac{x^{5-1}}{2} y^{-3-5} = \frac{x^{4}}{2} y^{-8}.$$\n\n3. **Apply the negative exponent:**\n\n$$\left( \frac{x^{4}}{2} y^{-8} \right)^{-4} = \left( \frac{x^{4} y^{-8}}{2} \right)^{-4} = \left( \frac{x^{4} y^{-8}}{2} \right)^{-4} = \left( \frac{2}{x^{4} y^{-8}} \right)^{4} = \frac{2^{4}}{x^{16} y^{-32}} = \frac{16}{x^{16} y^{-32}}.$$\n\n4. **Simplify the denominator of the division:**\n\nSimplify $$\left( 3x^{-2}y^{2} \right)^{-3} = 3^{-3} x^{6} y^{-6} = \frac{1}{27} x^{6} y^{-6}.$$\n\nThen, the denominator is\n$$\frac{4x^{6}y^{-10}}{\frac{1}{27} x^{6} y^{-6}} = 4x^{6} y^{-10} \times \frac{27}{x^{6} y^{-6}} = 4 \times 27 \times \frac{x^{6}}{x^{6}} \times \frac{y^{-10}}{y^{-6}} = 108 y^{-10+6} = 108 y^{-4}.$$\n\n5. **Divide the two results:**\n\n$$\frac{16}{x^{16} y^{-32}} \div 108 y^{-4} = \frac{16}{x^{16} y^{-32}} \times \frac{1}{108 y^{-4}} = \frac{16}{108 x^{16} y^{-32} y^{-4}} = \frac{16}{108 x^{16} y^{-36}}.$$\n\n6. **Simplify the fraction:**\n\n$$\frac{16}{108} = \frac{\cancel{16}^{4}}{\cancel{108}^{27}} = \frac{4}{27}.$$\n\n7. **Final simplified expression:**\n\n$$\frac{4}{27} x^{-16} y^{36} = \frac{4 y^{36}}{27 x^{16}}.$$\n\n**Answer:** $$\boxed{\frac{4 y^{36}}{27 x^{16}}}.$$