The age of a son today is ${\frac{1}{4}}^{\text{th}}$ of the age of his father. After $20$ years, the age of the father will be twice the age of his son. The age of the father today is:

Let the current age of the father be $F$ and the current age of the son be $S$.
Given:

$$S=\frac{1}{4}F⟹F=4S$$

After 20 years:

$$\text{Father’s age}=F+20$$

$$\text{Son’s age}=S+20$$

According to the condition:

$$F+20=2(S+20)$$

Substitute $F=4S$:

$$4S+20=2S+40$$

$$2S=20⟹S=10\text{ years}$$

Therefore, the current age of the father is:

$$F=4\times 10=40\text{ years}$$