The question relates to the concept of light reflection and optics, specifically about how light behaves when passing through media of different refractive indices.

1. The periscope utilizes two plane mirrors to redirect light, allowing the observer to see over obstacles. Normally, mirrors in a periscope are set at an angle of \(45^{\circ}\).
2. In this scenario, the tube of the periscope is filled with a transparent liquid with a refractive index of \(1.5\). This means light traveling through the liquid will experience refraction but will not change its path when it reflects off the mirrors.
3. The refractive index has no effect on the brightness or lateral position of the image as perceived through the periscope. The properties of the reflected light (such as intensity and direction) depend only on the reflective surfaces of the mirrors, not on the medium between them.
4. Let's evaluate the options:
   • The image will appear \(1.5\) times darker: This option is incorrect because the refractive index does not directly affect the brightness or intensity of the reflected light from the mirrors.
   • The image will appear laterally shifted: This is incorrect because a lateral shift would imply the image has moved sideways, which is not affected by a change in the medium.
   • The image will appear \(1.5\) times brighter: Again, this is incorrect because the medium’s refractive index does not change the brightness of the mirror-reflected image.
   • The image will remain unchanged: This is correct. In terms of appearance, the image will be unaffected by the refractive index change since reflection on plane mirrors does not depend on the medium but rather on the mirrors themselves.
5. Therefore, the correct interpretation is that the image will remain unchanged.