1. **State the problem:** Find the fourth term in the expansion of $\left(2x - \frac{1}{x}\right)^{10}$.\n\n2. **Formula used:** The binomial expansion of $(a + b)^n$ is given by $$\sum_{k=0}^n \binom{n}{k} a^{n-k} b^k.$$\n\n3. **Identify terms:** Here, $a = 2x$, $b = -\frac{1}{x}$, and $n=10$. The general term (the $(k+1)$th term) is $$T_{k+1} = \binom{10}{k} (2x)^{10-k} \left(-\frac{1}{x}\right)^k.$$\n\n4. **Find the fourth term:** The fourth term corresponds to $k=3$. Substitute $k=3$:\n$$T_4 = \binom{10}{3} (2x)^{7} \left(-\frac{1}{x}\right)^3.$$\n\n5. **Calculate binomial coefficient:** $$\binom{10}{3} = \frac{10!}{3!7!} = 120.$$\n\n6. **Simplify powers:**\n$$(2x)^7 = 2^7 x^7 = 128 x^7,$$\n$$\left(-\frac{1}{x}\right)^3 = (-1)^3 \frac{1}{x^3} = -\frac{1}{x^3}.$$\n\n7. **Combine terms:**\n$$T_4 = 120 \times 128 x^7 \times \left(-\frac{1}{x^3}\right) = 120 \times 128 \times (-1) x^{7-3} = -15360 x^4.$$\n\n**Final answer:** The fourth term in the expansion is $$\boxed{-15360 x^4}.$$