When the numerator of a fraction increases by $4$, the fraction increases by $\frac{2}{3}$. The denominator of the fraction is

Let the original fraction be $\frac{x}{y}$, where $x$ is the numerator and $y$ is the denominator.
According to the problem, when the numerator increases by $4$:

$$\frac{x+4}{y}=\frac{x}{y}+\frac{2}{3}$$

Splitting the fraction on the left-hand side:

$$\frac{x}{y}+\frac{4}{y}=\frac{x}{y}+\frac{2}{3}$$

Subtracting $\frac{x}{y}$ from both sides gives:

$$\frac{4}{y}=\frac{2}{3}$$

$$2y=12$$

$$y=6$$

Therefore, the denominator of the fraction is $6$.