If there is no clause or clauses that extend the primary term of the lease, the lease will expire at the end of the primary term.
It doesn’t make any difference if there is no drilling rig on site. The land may be unitized with other lands, and the drilling rig may be located elsewhere.
Once again, a survey plat and a drilling permit would typically be an action by lessee that is performed in the primary term of the lease. The question doesn’t really make sense in that regard.
It sounds like by having done additional research, the lessee believes that you are unitized. If that is the case, and they don’t want to pay for a new lease, ask them about royalty payments.
You seem to be fishing for an answer that isn’t there. Good luck to you.