in what conditions a charge may start a simple harmonic motion when kept in electric fields of other charges?

according to Earnshaws theorem, no charge can be kept in STABLE equilibrium using electrostatic fields only. In addition SHM cannot occur if the equilibrium is not stable. So the answer is: under no conditions.

Hi Priyanka,

This is possible only under the condition, that the charge is a "Time Varying Charge"

And hence the force F = qE, would vary with time and under suitable variations of the charge it can exhibit a Harmonic Motion.

(This is out of Scope of syllabus of IIT JEE and also your Boards)

And as mentioned by the above user, Earnshaws theorem, is applicable only on Charges that are time invariant.

So, for a constant charge, it would never be able to execute a Harmonic Motion.

All the best.

Regards,

Ashwin (IIT Madras).