1. Recall the definition of Young's modulus

   Young's modulus is a material property defined as $Y = \frac{\text{stress}}{\text{strain}} = \frac{\frac{F}{A}}{\frac{\Delta L}{L}} = \frac{FL}{A\,\Delta L}$

2. Key idea

   Young's modulus depends only on the nature of the material, not on the dimensions of the wire.

   So changing:

   • length from $L$ to $2L$
   • radius from $r$ to $\frac{r}{2}$

   does not change the Young's modulus of the material.

3. Conclusion

   Therefore, the Young's modulus will remain the same.

4. Option check

   • A: remain same ✅
   • B: become $8$ times its initial value ❌
   • C: become $\frac{1}{4}$ of its initial value ❌
   • D: become $4$ times its initial value ❌

Hence, the correct answer is A.