A particle remains at rest when observed from a certain frame. We...

If a particle is at rest in a given frame of reference, its acceleration with respect to that frame is zero.

If the frame is an inertial frame, the acceleration of the particle with respect to the ground is zero. According to Newton's second law, the net real force acting on the particle must be zero.

If the frame is a non-inertial frame accelerating with $\vec{a}_0$ with respect to the ground, the particle must also have an acceleration $\vec{a}_0$ with respect to the ground to remain at rest in the non-inertial frame. Thus, the net real force acting on the particle is $m\vec{a}_0$, which is non-zero.

Therefore, the frame might be inertial with zero resultant force, or it might be non-inertial with a non-zero resultant force.