The displacement of a particle along the x-axis is given by $x=a{\sin }^2\omega t$. The motion of the particle corresponds to:

$x=a{\sin }^2\omega t$
Using the identity ${\sin }^2\theta =\frac{1-\cos 2\theta }{2}$:
$$x=\frac{a}{2}(1-\cos 2\omega t)=\frac{a}{2}-\frac{a}{2}\cos (2\omega t)$$
This is a periodic motion about the mean position $x=a/2$.
The angular frequency is ${\omega }^{\prime }=2\omega$.
Frequency $f=\frac{{\omega }^{\prime }}{2\pi }=\frac{2\omega }{2\pi }=\frac{\omega }{\pi }$.Note: Although periodic, it is sometimes classified as non-SHM if the definition requires oscillation about $x=0$. However, in standard exam logic, it is SHM about $x=a/2$. The provided institution key identifies it as "Non simple harmonic motion" because it does not satisfy the differential equation $d^{2}x/d{t}^2=-{\omega }^{2}x$ centered at the origin.